Probabilistic morphisms and Bayesian nonparametrics

TL;DR

This paper develops a functorial framework for probabilistic morphisms and applies it to Bayesian nonparametrics, providing new insights into posterior distributions and Dirichlet measures within a categorical setting.

Contribution

It unifies existing categories of probabilistic morphisms and statistical models, introduces a formal Bayesian model with explicit posterior formulas, and offers a new proof for Dirichlet measures using functorial properties.

Findings

Explicit formula for posterior distributions in Bayesian models with specific conditions.

Unified categorical framework for probabilistic morphisms and statistical models.

New proof of the existence of Dirichlet measures using functorial properties.

Abstract

In this paper we develop a functorial language of probabilistic morphisms and apply it to some basic problems in Bayesian nonparametrics. First we extend and unify the Kleisli category of probabilistic morphisms proposed by Lawvere and Giry with the category of statistical models proposed by Chentsov and Morse-Sacksteder. Then we introduce the notion of a Bayesian statistical model that formalizes the notion of a parameter space with a given prior distribution in Bayesian statistics. {We revisit the existence of a posterior distribution, using probabilistic morphisms}. In particular, we give an explicit formula for posterior distributions of the Bayesian statistical model, assuming that the underlying parameter space is a Souslin space and the sample space is a subset in a complete connected finite dimensional Riemannian manifold. Then we give a new proof of the existence of Dirichlet measures over any measurable space using a functorial property of the Dirichlet map constructed by Sethuraman.

Equations

$${\mu }_{\Theta |\mathcal{X}}(B|\mathbf{x})=\underset{r\to 0}{\lim }\frac{{\int }_B{\Pi }_{i=1}^k{p}_{\theta }(D_r(x_i))d{\mu }_{\Theta }}{{\int }_{\Theta }{\Pi }_{i=1}^k{p}_{\theta }(D_r(x_i))d{\mu }_{\Theta }}$$

$$\mathcal{M}(\mathcal{X})={\cap }_{A\in {\mathcal{A}}_{\mathcal{X}}}⟨A,{\mathbb{R}}_{\ge 0}⟩,$$

$${\mathcal{M}}^{\ast }(\mathcal{X})=\mathcal{M}(\mathcal{X})\cap ⟨\mathcal{X},{\mathbb{R}}_{>0}⟩$$

$$\mathcal{P}(\mathcal{X})=\mathcal{M}(\mathcal{X})\cap ⟨\mathcal{X},1⟩$$

$$I_f\circ \mathfrak{a}(\mu ,\nu )=I_{1_{\mathcal{X}}}(f\cdot \mu )+I_{1_{\mathcal{X}}}(f\cdot \nu ),$$

$${\mu }_{\mathcal{Y}|\mathcal{X}}(y\in B|x):=\frac{d({\Pi }_{\mathcal{X}}{)}_{\ast }(1_{B\times \mathcal{X}}{\mu }_{\mathcal{X}\times \mathcal{Y}})}{d({\Pi }_{\mathcal{X}}{)}_{\ast }{\mu }_{\mathcal{X}\times \mathcal{Y}}}(x)$$

$$\overline{T}(x)(B):=\mathbb{P}\{\omega \in \Omega :T(\omega ,x)\in B\}=\mathbb{P}\{T^{-1}(\cdot ,x)(B)\},\forall B\in {\Sigma }_{\mathcal{Y}}.$$

$$T^{\ast }(f)(x):=I_f(\overline{T}(x))={\int }_{\mathcal{Y}}fd\overline{T}(x).$$

$$S_{\ast }(T)(\mu )(B):={\int }_{\mathcal{X}}\overline{T}(x)(B)d\mu (x)$$

$$e{v}_{\mu }:(L(\mathcal{X}),{\Sigma }_{ev})\to \mathbb{R},h\mapsto I_h(\mu )$$

$${\mu }^k=\sum _{i=1}^{\infty }\mu (B_{k,m}){\delta }_{x_{k,i}}$$

$$\mu (h)=\underset{k\to \infty }{\lim }{\mu }^k(h).$$

$$S_{\ast }(T_2\circ T_1)=S_{\ast }(T_2)\circ S_{\ast }(T_1),P_{\ast }(T_2\circ T_1)=P_{\ast }(T_2)\circ P_{\ast }(T_1).$$

$$\underset{x\in \mathcal{X}}{\sup }|T^{\ast }(f)(x)|\le \underset{y\in \mathcal{Y}}{\sup }|f(y)|$$

$$T^{\ast }(f{)}^{-1}(I)=\{x\in \mathcal{X}|T^{\ast }(f)(x)\in I,\text{ i.e., }{I}_f(\overline{T}(x))\in I\}.$$

$$e{v}_{\mu }^{-1}(T^{\ast }{)}^{-1}(I)=\{g\in L(\mathcal{Y})|{\int }_{\mathcal{X}}{\int }_{\mathcal{Y}}gd\overline{T}(x)d\mu (x)\in I\}.$$

$${\overline{T}}_{\mu }:={\int }_{\mathcal{X}}\overline{T}(x)d\mu \in \mathcal{P}(\mathcal{Y})$$

$$e{v}_{\mu }^{-1}(T^{\ast }{)}^{-1}(I)=e{v}_{{\overline{T}}_{\mu }}^{-1}(I)$$

$${\mathcal{P}}^2(\mathcal{X})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$e{v}_P({\nu }_{\mathcal{P}})(A):={\int }_{\mathcal{P}(\mathcal{X})}{I}_{1_A}(\mu )d{\nu }_{\mathcal{P}}(\mu )$$

$${\mathcal{P}}^2(\mathcal{X})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$P_{\ast }(T_3\circ T_2)=P_{\ast }(T_3)\circ P_{\ast }(T_2).$$

$$P_{\ast }(T)=e{v}_P\circ P_{\ast }(\overline{T}),$$

$$\overline{T_2\circ T_1}=P_{\ast }(T_2)\circ \overline{T_1}.$$

$$e{v}_P\circ P_{\ast }(\overline{T})(\mu )(B)={\int }_{\mathcal{P}(\mathcal{Y})}{I}_{1_B}d{P}_{\ast }(\overline{T})\mu ={\int }_{\mathcal{X}}\overline{T}(x)(B)d\mu (x)=P_{\ast }(T)(\mu )(B).$$

$$P_{\ast }(T_3\circ T_2)=e{v}_P\circ P_{\ast }(\overline{T_3\circ T_2})$$

$$=e{v}_P\circ P_{\ast }(P_{\ast }(T_3)\circ \overline{T_2})$$

$$=e{v}_P\circ P_{\ast }(e{v}_P)\circ P_{\ast }^2({\overline{T}}_3)\circ P_{\ast }({\overline{T}}_2).$$

$$e{v}_P\circ P_{\ast }(e{v}_P)=e{v}_P\circ e{v}_P.$$

$$e{v}_P\circ P_{\ast }^2(\overline{T})=P_{\ast }(\overline{T})\circ e{v}_P.$$

$$\overline{T_3\circ (T_2\circ T_1)}=\overline{(T_3\circ T_2)\circ T_1}.$$

Full text

1 Introduction

Statistical inference is a basic tool for essentially all sciences when one wants to detect some structure behind observed data from some measurable sample space ${\mathcal{X}}$ and make predictions about future observations. Therefore, many useful techniques for making inferences have been developed. Beyond this practical aspect, of course, one also seeks a deeper understanding of the underlying principles. Formal methods for that have been developed in various disciplines, including statistical physics, in particular, using tools from category theory. We shall therefore now briefly recall the history of the categorical approach to mathematical statistics, to the extent that it is relevant for our work.

When Lawvere Lawvere1962 developed a categorical approach to mathematical statistics, he introduced the name probabilistic mappings for Markov kernels. Lawvere and later Giry Giry1982 proved important functorial properties of probabilistic mappings. Almost at the same time Morse-Sacksteder MS1966 and Chentsov Chentsov1965 developed independently categorical approaches for mathematical statistics, namely the category of statistical models. The purpose of our paper is to unify their approaches and develop the functorial language of probabilistic mappings, renamed in this paper as probabilistic morphisms following a suggestion by Juan Pablo Vigneaux, for dealing with important problems in Bayesian nonparametrics. In particular, we introduce the notion of a Bayesian statistical model (Definition 5) and express the existence of posterior distributions of the Bayesian statistical model in terms of the equivalence between statistical models (Proposition 8), generalizing Parzygnat’s theorem (Parzygnat2020, , Theorem 2.1) for finite sets. We also prove a new formula for these posterior distributions without the dominated measure condition (Theorem 1.1).

Recall that a Souslin space is a Hausdorff space admitting a surjective continuous mapping from a complete separable metrizable space, i.e., a Polish space. In particular, every Polish space is a Souslin space, and more generally, every Borel subset in a Souslin space is a Souslin space (Bogachev2007, , Corolary 6.6.7, p. 22, vol.2).

Remark 1

1. A Souslin space need not be metrizable, see e.g. (Bogachev2007, , Example 6.10.78, p. 64, vol. 2).

2. Every topological space ${\mathcal{X}}$ has a structure of a measurable space, also denoted by ${\mathcal{X}}$, whose $\sigma$-algebra is the Borel $\sigma$-algebra ${\mathcal{B}}({\mathcal{X}})$. In (DM1978, , Definition 16, p.46-III) Dellacherie and Meyer called a measurable space $({\mathcal{X}},\Sigma_{{\mathcal{X}}})$ Souslin, if $({\mathcal{X}},\Sigma_{{\mathcal{X}}})$ is isomorphic to a measurable space $({\mathcal{H}},{\mathcal{B}}({\mathcal{H}}))$ where ${\mathcal{H}}$ is a Souslin metrizable space. Here, we call such an $({\mathcal{X}},\Sigma_{{\mathcal{X}}})$ a Souslin measurable space. In (DM1978, , Theorem 68, p. 76-III) Dellacherie and Meyer proved that, if ${\mathcal{X}}$ is a Souslin space, then $({\mathcal{X}},{\mathcal{B}}({\mathcal{X}}))$ is a Souslin measurable space.

3. Since any Borel subset in a Souslin space is a Souslin space, any standard Borel space (a measurable space admitting a bijective, bimeasurable correspondence with a Borel subset of a Polish space) is a Souslin measurable space.

Our main theorem is then stated as follows:

Theorem 1.1 (Main Theorem)

Suppose that ${\mathcal{X}}$ is a subset of a connected complete finite dimensional Riemannian manifold $(M^{n},g)$ with the induced metric generated by $g$ on $M^{n}$, and $(\Theta,\mu_{\Theta},{\mathbf{p}},{\mathcal{X}})$ is a Bayesian statistical model, where $\Theta$ is a Souslin space. For any $k\in{\mathbb{N}}^{+}$ and ${\bf x}=(x_{1},\cdots,x_{k})\in{\mathcal{X}}^{k}$ let $D_{r}(x_{i})$ denote the open ball of radius $r$ centered at $x_{i}\in{\mathcal{X}}$. For $\theta\in\Theta$ we let $p_{\theta}:={\mathbf{p}}(\theta)\in{\mathcal{P}}({\mathcal{X}})$. Let $\mu_{\mathcal{X}}:={\mathbf{p}}_{*}(\mu_{\Theta})\in{\mathcal{P}}({\mathcal{X}})$ be the marginal measure on ${\mathcal{X}}$. Then there exists a measurable subset $S\subset{\mathcal{X}}$ of zero $\mu_{\mathcal{X}}$-measure independent of ${\bf x}$ and a family of posterior distributions $\mu_{\Theta|{\mathcal{X}}}(\cdot|{\bf x})$ on $\Theta$ after seeing data ${\bf x}\in{\mathcal{X}}^{k}$ such that

[TABLE]

for any $B$ in the Borel $\sigma$-algebra ${\mathcal{B}}(\Theta)=\Sigma_{\Theta}$, and for any ${\bf x}\in({\mathcal{X}}\setminus S)^{k}$. For ${\bf x}\in{\mathcal{X}}^{k}\setminus({\mathcal{X}}\setminus S)^{k}$ we take an arbitrary value for $\mu_{\Theta|{\mathcal{X}}}(\cdot|{\bf x})\in{\mathcal{P}}(\Theta)$.

Furthermore, we show that the singular set $S$ in Theorem 1.1 remains a zero measure set with respect to the updated marginal measure (Proposition 14). Thus our formula is consistent for Bayes inference and can be applied to Bayesian nonparametrics when the dominated measure condition usually does not hold. See, e.g. GV2017 , for a discussion of problems in Bayesian nonparametrics when the assumption of the dominated measure condition is dropped off and therefore we cannot apply the classical Bayes formula.

Bayes statistics and the Bayes formula have been treated in several papers devoted to categorical approaches to probability and statistics, CJ2019 , Fritz2020 , Parzygnat2020 but the related results in the cited papers are “synthetic” as emphasized by Fritz Fritz2020 , in the sense that existence and uniqueness theorems therein are formal/abstract, in contrast to our Main Theorem, which is “analytic”, in the sense that our formula is explicit. To our knowledge, such an explicit Bayes formula did not exist in the classical literature on probability and mathematical statistics, except for the classical Bayes formula under the assumption of dominating measures (Schervish1997, , Theorem 1.31, p. 16), see also (53).

In our paper we also provide many examples to demonstrate the advantage of the functorial language of probabilistic morphisms. In particular we give a short functorial proof of the existence of Dirichlet measures (also known as Dirichlet processes) over any measurable space using the Sethuraman construction. Thus our functorial proof adds a demonstration of the advantage of the functorial language of probabilistic morphisms. We would like to refer the reader to Fritz2020 for a generalization of the category of probabilistic morphisms and their applications, in particular, Fritz’s reconsideration of several classical results in mathematical statistics, using the categorical approach. We refer the reader to Le2020 for further applications of probabilistic morphisms to the category of statistical models and their natural geometry.

To give an overview of the remainder of our paper let us fix some notations. For a measurable space ${\mathcal{X}}$, which sometimes we shall denote by $({\mathcal{X}},\Sigma_{\mathcal{X}})$ if we wish to specify the underlying $\sigma$-algebra $\Sigma_{\mathcal{X}}$, let ${\mathcal{S}}({\mathcal{X}})$, ${\mathcal{M}}({\mathcal{X}})$ and ${\mathcal{P}}({\mathcal{X}})$ denote the space of finite signed measures on ${\mathcal{X}}$, the space of non-negative measures and the space of probability measures on ${\mathcal{X}}$, respectively. We also set ${\mathcal{M}}^{*}({\mathcal{X}}):={\mathcal{M}}({\mathcal{X}})\setminus\{0\}$ and denote by $L({\mathcal{X}})$ 222In (Chentsov1972, , p. 74) Chentsov used the notation $L({\mathcal{X}},\Sigma_{\mathcal{X}})$ which is equivalent to our notation, and in (Bogachev2007, , p. 371, vol. 2) Bogachev used the notation ${\mathcal{L}}^{\infty}_{\Sigma_{\mathcal{X}}}$ instead of our notation $L({\mathcal{X}})$. the space of bounded measurable functions on ${\mathcal{X}}$. Furthermore, we denote by $1_{A}$ the indicator (characteristic) function of a measurable set $A$. For a $\sigma$-additive measure $\mu$ we denote by $\mu^{*}$ its outer measure.

In the second section we extend the Kleisli category of probabilistic morphisms proposed by Lawvere Lawvere1962 and Giry Giry1982 and unify it with the category of statistical models proposed by Chentsov Chentsov1965 , Chentsov1972 and by Morse-Sacksteder MS1966 . In the third section, using the results in the second section, we introduce the notion of a Bayesian statistical model that formalizes the notion of a parameter space with a prior distribution in Bayesian statistics. Then we express the existence of a posterior distribution using functorial language and give a formula for the posterior distribution without the dominated measure model condition. This will depend on precise measure theoretical arguments, including a version of the Besicovich covering lemma for Riemannian manifolds proved in (Federer1969, , 2.8.9), see also JLT2020 . In the fourth section we give a new proof of the existence of Dirichlet measures over any measurable space using a functorial property of the Dirichlet map constructed by Sethuraman.

2 Probabilistic morphisms and category of statistical models

In this section, first, we extend Lawvere’s natural $\sigma$-algebra on ${\mathcal{P}}({\mathcal{X}})$ to the spaces ${\mathcal{S}}({\mathcal{X}})$, ${\mathcal{M}}({\mathcal{X}})$, ${\mathcal{M}}^{*}({\mathcal{X}})$, $L({\mathcal{X}})$. We prove their important properties (Propositions 1, 2, 3, 4) that shall be needed in later sections. Then we extend related results due to Lawvere, Chentsov, Giry and Morse-Sacksteder (Theorem 2.1, Remark 6) concerning the Kleisli category of probabilistic mappings and show that the concept of the category of statistical models introduced by Chentsov and Morse-Sacksteder fits into the category of probabilistic morphisms (Proposition 7, Theorem 2.2, Remark 10).

2.1 Natural $\sigma$-algebras on ${\mathcal{S}}({\mathcal{X}})$, ${\mathcal{M}}({\mathcal{X}})$, ${\mathcal{P}}({\mathcal{X}})$ and $L({\mathcal{X}})$

Let ${\mathcal{X}}$ be a measurable space. Since we want to integrate not only on ${\mathcal{X}}$ itself, but also on spaces of measures on ${\mathcal{X}}$, we need to construct $\sigma$-algebras on those spaces. For that purpose, let ${\mathcal{F}}_{s}({\mathcal{X}})$ denote the linear space of simple (step) functions on ${\mathcal{X}}$. Such a function is a finite linear combination of indicator functions of measurable subsets of ${\mathcal{X}}$, and therefore, the space ${\mathcal{F}}_{s}({\mathcal{X}})$ depends on the $\sigma$-algebra $\Sigma_{\mathcal{X}}$. We shall now use this space to construct induced $\sigma$-algebras on our spaces of measures on ${\mathcal{X}}$. There is a natural homomorphism $I:{\mathcal{F}}_{s}({\mathcal{X}})\to{\mathcal{S}}^{*}({\mathcal{X}}):=Hom(S({\mathcal{X}}),{\mathbb{R}}),\,f\mapsto I_{f}$, defined by integration: $I_{f}(\mu):=\int_{\mathcal{X}}fd\mu$ for $f\in{\mathcal{F}}_{s}({\mathcal{X}})$ and $\mu\in{\mathcal{S}}({\mathcal{X}})$. Following Lawvere Lawvere1962 , we shall denote by $\Sigma_{w}$ the smallest $\sigma$-algebra on ${\mathcal{S}}({\mathcal{X}})$ such that $I_{f}$ is measurable for all $f\in{\mathcal{F}}_{s}({\mathcal{X}})$. We also denote by $\Sigma_{w}$ the restriction of $\Sigma_{w}$ to ${\mathcal{M}}({\mathcal{X}})$, ${\mathcal{M}}^{*}({\mathcal{X}})$ and ${\mathcal{P}}({\mathcal{X}})$. Since $I_{f}:{\mathcal{P}}({\mathcal{X}})\to{\mathbb{R}}$ is bounded, if $f$ is bounded, by the Lebesgue dominated convergence theorem, the $\sigma$-algebra $\Sigma_{w}$ on ${\mathcal{P}}({\mathcal{X}})$ is the smallest $\sigma$-algebra on ${\mathcal{P}}({\mathcal{X}})$ such that $I_{f}$ is measurable for all $f\in L({\mathcal{X}})$.

Remark 2

Lawvere Lawvere1962 and later authors GH1989 , Kallenberg2017 333Kallenberg in (Kallenberg2017, , p.1) defined $\Sigma_{w}$ on the space of all locally finite measures on ${\mathcal{X}}$ as in Lawvere1962 . defined the $\sigma$-algebra on ${\mathcal{P}}({\mathcal{X}})$ as the smallest $\sigma$-algebra for which the evaluation map $ev_{A}:{\mathcal{P}}\to[0,1],\,\mu\mapsto\mu(A),$ is measurable for all $A\in\Sigma_{\mathcal{X}}$. It is not hard to see that their definition is equivalent to ours, since $\{I_{1_{A}}|\,A\in\Sigma_{\mathcal{X}}\}$ generate the vector space ${\mathcal{F}}_{s}$. The latter space leads directly to the space $L({\mathcal{X}})$.

For a topological space ${\mathcal{X}}$ we shall take as $\Sigma_{\mathcal{X}}$ the natural Borel $\sigma$-algebra ${\mathcal{B}}({\mathcal{X}})$, unless otherwise specified. Let $C_{b}({\mathcal{X}})\subset L({\mathcal{X}})$ be the space of bounded continuous functions on ${\mathcal{X}}$. We denote by $\tau_{w}$ the smallest topology on ${\mathcal{S}}({\mathcal{X}})$ such that for any $f\in C_{b}({\mathcal{X}})$ the map $I_{f}:({\mathcal{S}}({\mathcal{X}}),\tau_{w})\to{\mathbb{R}}$ is continuous. We also denote by $\tau_{w}$ the restriction of $\tau_{w}$ to ${\mathcal{M}}({\mathcal{X}})$ and ${\mathcal{P}}({\mathcal{X}})$, which is also called the weak topology. It is known that $({\mathcal{P}}({\mathcal{X}}),\tau_{w})$ is separable, metrizable if and only if ${\mathcal{X}}$ is (Bogachev2018, , Theorem 3.1.4, p. 104), (Parthasarathy1967, , Theorem 6.2, p.43) 444If ${\mathcal{X}}$ is infinite, then $({\mathcal{S}}({\mathcal{X}}),\tau_{w})$ is non-metrizable (Bogachev2018, , p. 102) (warning: Bogachev’s ${\mathcal{M}}({\mathcal{X}})$ is our ${\mathcal{S}}({\mathcal{X}})$).. If ${\mathcal{X}}$ is separable and metrizable then the Borel $\sigma$-algebra on ${\mathcal{P}}({\mathcal{X}})$ generated by $\tau_{w}$ coincides with $\Sigma_{w}$ (GH1989, , Theorem 2.3).

Proposition 1

(1) Assume that $\Sigma_{\mathcal{X}}$ has a countable generating algebra ${\mathcal{A}}_{\mathcal{X}}$. Then ${\mathcal{M}}({\mathcal{X}})$ is a measurable subset of ${\mathcal{S}}({\mathcal{X}})$, and ${\mathcal{P}}({\mathcal{X}})$ and ${\mathcal{M}}^{*}({\mathcal{X}})$ are measurable subsets of ${\mathcal{M}}({\mathcal{X}})$.

(2) The addition ${\mathfrak{a}}:({\mathcal{M}}({\mathcal{X}})\times{\mathcal{M}}({\mathcal{X}}),\Sigma_{w}\otimes\Sigma_{w})\to({\mathcal{M}}({\mathcal{X}}),\Sigma_{w}),\>(\mu,\nu)\mapsto\mu+\nu$, is a measurable map. If ${\mathcal{X}}$ is a topological space, then the map ${\mathfrak{a}}$ is $\tau_{w}$-continuous, i.e., continuous in the $\tau_{w}$-topology.

Proof

The $\sigma$-algebra $\Sigma_{w}$ on ${\mathcal{S}}({\mathcal{X}})$ is generated by subsets $\langle A,B^{*}\rangle:=I^{-1}_{1_{A}}(B^{*})$ where $A\in\Sigma_{\mathcal{X}}$ and $B^{*}\in{\mathcal{B}}({\mathbb{R}})$. We have

[TABLE]

because an element of ${\mathcal{M}}({\mathcal{X}})$ has to be nonnegative on every $A\in{\mathcal{A}}_{\mathcal{X}}$. When $\Sigma_{\mathcal{X}}$ has a countable generating algebra ${\mathcal{A}}_{\mathcal{X}}$, this is a countable intersection, implying the measurability of ${\mathcal{M}}({\mathcal{X}})$. And since

[TABLE]

and

[TABLE]

we then also obtain the measurability of ${\mathcal{M}}^{*}({\mathcal{X}})$ and ${\mathcal{P}}({\mathcal{X}})$. This proves (1).

2. To prove the measurability of the map ${\mathfrak{a}}$ it suffices to show that for any $f\in{\mathcal{F}}_{s}({\mathcal{X}})$ the composition $I_{f}\circ{\mathfrak{a}}:{\mathcal{M}}({\mathcal{X}})\times{\mathcal{M}}({\mathcal{X}})\to{\mathbb{R}}_{\geq 0}$ is measurable. Using the formula

[TABLE]

we reduce the measurability of $I_{f}\circ{\mathfrak{a}}$ to the measurability of the map ${\mathfrak{a}}:{\mathbb{R}}_{\geq 0}\times{\mathbb{R}}_{\geq 0}\to{\mathbb{R}}_{\geq 0},(x,y)\mapsto(x+y),$ which is well-known.

Similarly we prove the continuity of the map ${\mathfrak{a}}$, if ${\mathcal{X}}$ is a topological space. This proves Proposition 1(2).

Since the inclusions ${\mathcal{M}}({\mathcal{X}})\to{\mathcal{S}}({\mathcal{X}})$, ${\mathcal{M}}^{*}({\mathcal{X}})\to{\mathcal{S}}({\mathcal{X}})$ and ${\mathcal{P}}({\mathcal{X}})\to{\mathcal{S}}({\mathcal{X}})$) are measurable maps, any measure on ${\mathcal{M}}({\mathcal{X}})$, on ${\mathcal{M}}^{*}({\mathcal{X}})$ and on ${\mathcal{P}}({\mathcal{X}})$ is obtained by restricting a measure on ${\mathcal{S}}({\mathcal{X}})$ to ${\mathcal{M}}({\mathcal{X}})$, ${\mathcal{M}}^{*}({\mathcal{X}})$ and ${\mathcal{P}}({\mathcal{X}})$, respectively. By Proposition 1 the restriction of any measure on ${\mathcal{S}}({\mathcal{X}})$ to ${\mathcal{M}}({\mathcal{X}})$, ${\mathcal{M}}^{*}({\mathcal{X}})$ and ${\mathcal{P}}^{*}({\mathcal{X}})$) is a measure on ${\mathcal{M}}({\mathcal{X}})$, ${\mathcal{M}}^{*}({\mathcal{X}})$ and ${\mathcal{P}}^{*}({\mathcal{X}})$, respectively if $\Sigma_{\mathcal{X}}$ is countably generated.

2.2 Probabilistic morphisms and associated functors

Definition 1

A probabilistic morphism from a measurable space ${\mathcal{X}}$ to a measurable space ${\mathcal{Y}}$ (or an arrow from ${\mathcal{X}}$ to ${\mathcal{Y}}$) is a measurable mapping from ${\mathcal{X}}$ to $({\mathcal{P}}({\mathcal{Y}}),\Sigma_{w})$.

Remark 3

Definition 1 agrees with the definition of morphisms in the Kleisli category of the Giry probability monad, see Theorem 2.1 and Remark 6 below. (At this stage we have not defined the category where a probabilistic morphism is a morphism). Furthermore we would like to emphasize that a probabilistic morphism from ${\mathcal{X}}\leadsto{\mathcal{Y}}$ is not a mapping from ${\mathcal{X}}$ to ${\mathcal{Y}}$ as it assigns to $x\in{\mathcal{X}}$ not a single point $y\in{\mathcal{Y}}$, but rather a probability distribution on ${\mathcal{Y}}$. A probabilistic morphism is the same as a Markov kernel, see also Remark 6.

We shall denote by $\overline{T}:{\mathcal{X}}\to({\mathcal{P}}({\mathcal{Y}}),\Sigma_{w})$ the measurable mapping defining/generating a probabilistic morphism $T:{\mathcal{X}}\leadsto{\mathcal{Y}}$. Similarly, for a measurable mapping ${\mathbf{p}}:{\mathcal{X}}\to{\mathcal{P}}({\mathcal{Y}})$ we shall denote by $\underline{{\mathbf{p}}}:{\mathcal{X}}\leadsto{\mathcal{Y}}$ the generated probabilistic morphism. Note that a probabilistic morphism is denoted by a curved arrow and a measurable mapping by a straight arrow.

Example 1

(1) Assume that ${\mathcal{X}}$ is separable and metrizable. Then the identity mapping $Id_{\mathcal{P}}:({\mathcal{P}}({\mathcal{X}}),\tau_{w}w\to({\mathcal{P}}({\mathcal{X}}),\tau_{w})$ is continuous, and hence measurable w.r.t. the Borel $\sigma$-algebra $\Sigma_{w}={\mathcal{B}}(\tau_{w})$. Consequently $Id_{\mathcal{P}}$ generates a probabilistic morphism $ev:({\mathcal{P}}({\mathcal{X}}),{\mathcal{B}}(\tau_{w}))\leadsto({\mathcal{X}},{\mathcal{B}}({\mathcal{X}}))$ and we write $\overline{ev}=Id_{\mathcal{P}}$. Similarly, for any measurable space ${\mathcal{X}}$, we also have a probabilistic morphism $ev:({\mathcal{P}}({\mathcal{X}}),\Sigma_{w})\leadsto{\mathcal{X}}$ generated by the measurable mapping $\overline{ev}=Id_{\mathcal{P}}$.

(2) In generative models of supervised learning, we wish to know the stochastic correlation between a label $y$ in a sample space ${\mathcal{Y}}$ and an input $x$ in a sample space ${\mathcal{X}}$ that expresses the uncertain nature of the relationship between an input and its label. Such a stochastic correlation is expressed via a probability measure $\mu_{{\mathcal{X}}\times{\mathcal{Y}}}\in{\mathcal{P}}({\mathcal{X}}\times{\mathcal{Y}})$, assuming usually $\Sigma_{{\mathcal{X}}\times{\mathcal{Y}}}=\Sigma_{\mathcal{X}}\otimes\Sigma_{\mathcal{Y}}$. Denote by $\Pi_{\mathcal{X}}$ and $\Pi_{\mathcal{Y}}$ the natural projections from ${\mathcal{X}}\times{\mathcal{Y}}$ to ${\mathcal{X}}$ and ${\mathcal{Y}}$, respectively. One is then interested in the conditional probability

[TABLE]

for $B\in\Sigma_{\mathcal{Y}}$ and $x\in{\mathcal{X}}$. Here the equality (2) should be understood as an equivalence class of functions in $L^{1}({\mathcal{X}},(\Pi_{\mathcal{X}})_{*}\mu_{{\mathcal{X}}\times{\mathcal{Y}}})$. For simplification and practical purposes, we assume that the conditional probability $\mu_{{\mathcal{Y}}|{\mathcal{X}}}(\cdot|x)$ is regular, i.e., there is a function on ${\mathcal{X}}$ that represents the equivalence class of $\mu_{{\mathcal{Y}}|{\mathcal{X}}}(\cdot|x)$, and therefore we shall denote it also by $\mu_{{\mathcal{Y}}|{\mathcal{X}}}(\cdot|x)$ abusing notation, such that $\mu_{{\mathcal{Y}}|{\mathcal{X}}}(\cdot|x)$ is a measure on ${\mathcal{Y}}$ for all $x\in{\mathcal{X}}$, and the function $x\mapsto\mu_{{\mathcal{Y}}|{\mathcal{X}}}(B|x)$ is measurable for every $B\in\Sigma_{\mathcal{Y}}$. In this case the map $\overline{T}:{\mathcal{X}}\to{\mathcal{P}}({\mathcal{Y}}),x\mapsto\mu_{{\mathcal{Y}}|{\mathcal{X}}}(\cdot|x)$, is a measurable map. Thus under this regularity condition the problem of supervised learning with generative models is equivalent to the problem of finding a probabilistic morphism $T:{\mathcal{X}}\leadsto{\mathcal{Y}}$ describing the correlation between labeled data $y$ and input data $x$.

(3) We would like to present here a non-trivial construction of probabilistic morphisms, which comes from the theory of random mappings Kifer1986 , Kifer1988 . Consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a random mapping $T$, which is a measurable mapping $T:\Omega\times{\mathcal{X}}\to{\mathcal{Y}}$ (in case ${\mathcal{X}}={\mathcal{Y}}$, the random mapping $T$ is the source to generate a random dynamical system Arnold1998 ).

Such a random mapping $T:\Omega\times{\mathcal{X}}\to{\mathcal{Y}}$ then generates a measurable mapping $\overline{T}:{\mathcal{X}}\to{\mathcal{P}}({\mathcal{Y}})$, defined by

[TABLE]

Relation (3) shows that once a probabilistic sample space $(\Omega,\mathcal{F},\mathbb{P})$ is given, any random mapping would generate a probabilistic morphism. However, since probabilistic morphisms need not be constructed this way, the concept of probabilistic morphisms can be seen as a generalization of random mappings.

We are also interested in the inverse problem on the representation of probability measures, which can be formulated as follows: Given a probabilistic morphism $T:{\mathcal{X}}\leadsto{\mathcal{Y}}$, is there a probabilistic sample space $(\Omega,\mathcal{F},\mathbb{P})$ such that $T$ can be written as a random mapping from $\Omega\times{\mathcal{X}}$ to ${\mathcal{Y}}$? This question appears in several contexts in Ergodic Theory Kifer1986 , Kifer1988 . The answer is affirmative if ${\mathcal{Y}}$ is a compact, oriented, connected manifold of class $C^{2}$ and ${\mathcal{X}}$ is a $C^{0}$ manifold with some additional assumptions on the decay of the densities, where one can choose $\Omega:={\mathcal{Y}}$ (see details in (JMPR2019, , Theorem A)).

Given a probabilistic morphism $T:{\mathcal{X}}\leadsto{\mathcal{Y}}$, we define a linear map $T^{*}:L({\mathcal{Y}})\to L({\mathcal{X}})$ as follows

[TABLE]

Formula (4) coincides with the classical formula (5.1) in (Chentsov1972, , p. 66) for the transformation of a bounded measurable function $f$ under a Markov morphism $T$.555Chentsov called the induced transformation $P_{*}(T)$ (see Remark 4(2)) of a probabilistic morphism $T$ a Markov morphism Chentsov1965 . In particular, if $\kappa:{\mathcal{X}}\to{\mathcal{Y}}$ is a measurable mapping, then we have $\kappa^{*}(f)(x)=f(\kappa(x))$, since $\overline{\kappa}=\delta\circ\kappa$.

Further, we define a linear map $S_{*}(T):{\mathcal{S}}({\mathcal{X}})\to{\mathcal{S}}({\mathcal{Y}})$ as follows (Chentsov1972, , Lemma 5.9, p. 72)

[TABLE]

for any $\mu\in{\mathcal{S}}({\mathcal{X}})$ and $B\in\Sigma_{\mathcal{Y}}$.

Remark 4

(1) If $\kappa:{\mathcal{X}}\to{\mathcal{Y}}$ is a measurable mapping then $\kappa^{*}({\mathcal{F}}_{s}({\mathcal{Y}}))\subset{\mathcal{F}}_{s}({\mathcal{X}})$. For a general probabilistic morphism $T:{\mathcal{X}}\leadsto{\mathcal{Y}}$, we don’t have the inclusion $T^{*}({\mathcal{F}}_{s}({\mathcal{Y}}))\subset{\mathcal{F}}_{s}({\mathcal{X}})$, since the cardinality $\#(T^{*}(1_{B})(x)|\,x\in{\mathcal{X}})$ can be infinite. For example, consider $T=ev:{\mathcal{X}}:={\mathcal{P}}({\mathcal{Z}})\leadsto{\mathcal{Y}}:={\mathcal{Z}}$ for some measurable space ${\mathcal{Z}}$ and $\emptyset\not=B\not={\mathcal{Z}}$. Then $ev^{*}(1_{B})(\mu)=\mu(B)$ for any $\mu\in{\mathcal{X}}={\mathcal{P}}({\mathcal{Z}})$. Therefore $\#(ev^{*}(1_{B})(\mu)|\,\mu\in{\mathcal{P}}({\mathcal{Z}}))=\#(\mu(B)|\mu\in{\mathcal{P}}({\mathcal{Z}}))=\infty$, if ${\mathcal{Z}}=[0,1]$ and $B=[0,1/2]$.

(2) Lawvere also defined $P_{*}(T):{\mathcal{P}}({\mathcal{X}})\to{\mathcal{P}}({\mathcal{Y}})$ by (5) for $\mu\in{\mathcal{P}}({\mathcal{X}})$.

Definition 2

• We denote by $\Sigma_{*}$ the smallest $\sigma$-algebra on $L({\mathcal{X}})$ such that for any $\mu\in{\mathcal{P}}({\mathcal{X}})$ the evaluation function $ev_{\mu}:L({\mathcal{X}})\to{\mathbb{R}},h\mapsto I_{h}(\mu)$, is measurable.

• We denote by $\Sigma_{ev}$ the smallest $\sigma$-algebra on $L({\mathcal{X}})$ such that for all $x\in{\mathcal{X}}$ the evaluation map $ev_{x}:L({\mathcal{X}})\to{\mathbb{R}},h\mapsto h(x),$ is measurable.

Example 2

Identifying $L({\Omega}_{k})=C_{b}({\Omega}_{k})$ with ${\mathcal{S}}({\Omega}_{k})$ via the Euclidean metric, and noting that the $\sigma$-algebra $\Sigma_{w}$ on ${\mathcal{S}}({\Omega}_{k})$ is the usual Borel $\sigma$-algebra on ${\mathbb{R}}^{k}={\mathcal{S}}({\Omega}_{k})$, cf. (GH1989, , Theorem 2.3), it is not hard to see that the $\sigma$-algebra $\Sigma_{*}$ on $L({\Omega}_{k})={\mathbb{R}}^{k}$ coincides with the Borel $\sigma$-algebra on ${\mathbb{R}}^{k}$.

Remark 5

(1) For any $x\in{\mathcal{X}}$ denote by $\delta_{x}\in{\mathcal{P}}({\mathcal{X}})$ the Dirac measure concentrated at $x$. Giry proved that the map $\delta:{\mathcal{X}}\to({\mathcal{P}}({\mathcal{X}}),\Sigma_{w}),x\mapsto\delta_{x},$ is a measurable mappingGiry1982 . Furthermore, for $x\in{\mathcal{X}}$ and $h\in L({\mathcal{X}})$ we have $ev_{\delta(x)}(h)=ev_{x}(h)$.

(2) The requirement that the evaluation maps $ev_{x}$ are measurable is natural in the theory of measurable mappings Aumann1961 with applications in stochastic processes Baudoin2014 .

Proposition 2

For any measurable space ${\mathcal{X}}$ we have $\Sigma_{ev}\subset\Sigma_{*}$. If ${\mathcal{X}}$ is a separable metrizable space then $\Sigma_{ev}=\Sigma_{*}$.

Proof

Let ${\mathcal{X}}$ be a measurable space and $x\in{\mathcal{X}}$. Since the restriction of the evaluation mapping $ev_{x}$ to $L({\mathcal{X}})$ is equal to the evaluation mapping $ev_{\delta(x)}$, we obtain immediately the first assertion of Proposition 2.

To prove the second assertion it suffices to show that the evaluation mapping

[TABLE]

is measurable for any $\mu\in{\mathcal{P}}({\mathcal{X}})$. Since ${\mathcal{X}}$ is a separable metrizable space, by (Bogachev2018, , Example 2.2.7, p. 55), for any $k\in{\mathbb{N}}$ there exist a partition of ${\mathcal{X}}$ in countably many Borel parts, ${\mathcal{X}}=\cup_{i=1}^{\infty}B_{k,i}$ and a sequence of points $x_{k,i}\in B_{k,i}$ such that the sequence

[TABLE]

converges weakly to $\mu$.

Then for any $h\in L({\mathcal{X}})$ we have

[TABLE]

Since $ev_{\mu^{k}}:(L({\mathcal{X}}),\Sigma_{ev})\to{\mathbb{R}}$ is measurable, it follows that $ev_{\mu}:(L({\mathcal{X}}),\Sigma_{ev})\to{\mathbb{R}}$ is measurable. This completes the proof of Proposition 2.

Proposition 3

Assume that $T:{\mathcal{X}}\leadsto{\mathcal{Y}}$ is a probabilistic morphism.

(1) Then $T$ induces a bounded linear map $S_{*}(T):{\mathcal{S}}({\mathcal{X}})\to{\mathcal{S}}({\mathcal{Y}})$ w.r.t. the total variation norm $||\cdot||_{TV}$. The restriction $M_{*}(T)$ of $S_{*}(T)$ to ${\mathcal{M}}({\mathcal{X}})$ and $P_{*}(T)$ of $S_{*}(T)$ to ${\mathcal{P}}({\mathcal{X}})$, maps ${\mathcal{M}}({\mathcal{X}})$ to ${\mathcal{M}}({\mathcal{Y}})$ and ${\mathcal{P}}({\mathcal{X}})$ to ${\mathcal{P}}({\mathcal{Y}})$, respectively.

(2) If $T$ is a measurable mapping, then $S_{*}(T):({\mathcal{S}}({\mathcal{X}}),\Sigma_{w})\to({\mathcal{S}}({\mathcal{Y}}),\Sigma_{w})$ is a measurable map. It is $\tau_{w}$-continuous if $T$ is a continuous map between separable metrizable spaces. Hence the maps $M_{*}(T)$ and $P_{*}(T)$ are measurable and they are $\tau_{w}$-continuous if $T$ is a continuous map between separable metrizable spaces.

(3) If $T_{1}:{\mathcal{X}}\to{\mathcal{Y}}$ and $T_{2}:{\mathcal{Y}}\to{\mathcal{Z}}$ are measurable mappings then we have

[TABLE]

(4) The map $T$ also induces measurable linear mappings $T^{*}:(L({\mathcal{Y}}),\Sigma_{*})\to(L({\mathcal{X}}),\Sigma_{*})$ and $T^{*}:(L({\mathcal{Y}}),\Sigma_{ev})\to(L({\mathcal{X}}),\Sigma_{ev})$.

Proof

1. Proposition 3 (1) is due to Chentsov (Chentsov1972, , Lemma 5.9, p.72).

2. Assume that $T$ is a measurable mapping. To prove that $S_{*}(T)$ is a measurable mapping, it suffices to show that for any $f\in{\mathcal{F}}_{s}({\mathcal{Y}})$ the composition $I_{f}\circ S_{*}(T):{\mathcal{S}}({\mathcal{X}})\to{\mathbb{R}}$ is measurable. The latter assertion follows from the identity $I_{f}\circ S_{*}(T)=I_{T^{*}f}$, taking into account that $T^{*}(f)\in{\mathcal{F}}_{s}({\mathcal{X}})$. In the same way we prove that $S_{*}(T)$ is $\tau_{w}$-continuous, if $T$ is continuous map between separable metrizable spaces. The statement concerning $M_{*}(T)$ and $S_{*}(T)$ is a consequence of the first two assertions of Proposition 3(2). Note that the measurability of $P_{*}(T)$ has been first noticed by Lawvere Lawvere1962 .

3. The first identity in (6) is obvious, the second one is a consequence of the first one.

4. The linearity of $T^{*}$ and the inclusion $T^{*}(L({\mathcal{Y}}))\subset L({\mathcal{X}})$ have been proved in (Chentsov1972, , Corollary, p. 66). Here we provide an alternative shorter proof of the inclusion relation. Let $T^{*}$ be defined by (4) and $f\in L({\mathcal{Y}})$. Since

[TABLE]

the function $T^{*}(f)$ is bounded on ${\mathcal{X}}$.

Next we shall show that for any $f\in L({\mathcal{Y}})$ the map $T^{*}(f):{\mathcal{X}}\to{\mathbb{R}}$ is measurable, i.e., for any Borel set $I\subset{\mathbb{R}}$ we have $T^{*}(f)^{-1}(I)\in\Sigma_{\mathcal{X}}$. Note that

[TABLE]

Since $I_{f}:{\mathcal{P}}({\mathcal{Y}})\to{\mathbb{R}}$ is measurable, the set $I_{f}^{-1}(I)$ is measurable. Since $\overline{T}:{\mathcal{X}}\to{\mathcal{P}}({\mathcal{Y}})$ is measurable, the set $(T^{*}(f))^{-1}(I)=\overline{T}^{-1}(I_{f}^{-1}(I))$ is measurable. Thus $T^{*}(L({\mathcal{Y}}))\subset L({\mathcal{X}})$.

Now we shall show that $T^{*}:(L({\mathcal{Y}}),\Sigma_{*})\to(L({\mathcal{X}}),\Sigma_{*})$ is a measurable map. It suffices to show that for any $\mu\in{\mathcal{P}}({\mathcal{X}})$ the composition $ev_{\mu}\circ T^{*}:L({\mathcal{Y}})\to{\mathbb{R}}$ is a measurable map. Let $I\subset{\mathbb{R}}$ be a Borel subset. Then

[TABLE]

Setting

[TABLE]

we get

[TABLE]

which is a measurable subset of $L({\mathcal{Y}})$. Hence $T^{*}:(L({\mathcal{Y}}),\Sigma_{*})\to(L({\mathcal{X}}),\Sigma_{*})$ is a measurable map.

From (7) we also deduce that $T^{*}:(L({\mathcal{Y}}),\Sigma_{ev})\to(L({\mathcal{X}}),\Sigma_{ev})$ is a measurable map, since $ev_{x}(h)=ev_{\delta(x)}(h)$ for all $x\in{\mathcal{X}}$ and $h\in L({\mathcal{X}})$. This completes the proof of Proposition 3.

Proposition 4

(1) The projection $\pi_{\mathcal{X}}:{\mathcal{M}}^{*}({\mathcal{X}})\to{\mathcal{P}}({\mathcal{X}}),\mu\mapsto\mu({\mathcal{X}})^{-1}\cdot\mu$, is a measurable retraction. If ${\mathcal{X}}$ is a topological space then $\pi_{\mathcal{X}}$ is $\tau_{w}$-continuous.

(2) If $\Sigma_{\mathcal{X}}$ is countably generated, then we have the following commutative diagram

[TABLE]

where all arrows are measurable embeddings. If ${\mathcal{X}}$ is a separable metrizable space then the arrows are $\tau_{w}$-continuous embeddings.

Proof

1. Clearly $\pi_{\mathcal{X}}$ is a retraction. The map $\pi_{\mathcal{X}}$ is measurable, since the mapping $I_{1_{\mathcal{X}}}:{\mathcal{M}}^{*}({\mathcal{X}})\to{\mathbb{R}}_{>0},\>\mu\mapsto\mu({\mathcal{X}}),$ is measurable, and hence the map $(I_{1_{\mathcal{X}}})_{-1}:{\mathcal{M}}^{*}({\mathcal{X}})\to{\mathbb{R}}_{>0},\>\mu\mapsto\mu({\mathcal{X}})^{-1},$ is also measurable. Similarly we prove that $\pi_{\mathcal{X}}$ is continuous in the $\tau_{w}$-topology, if ${\mathcal{X}}$, ${\mathcal{Y}}$ are topological spaces.

2. The measurability of the horizontal mappings $i_{p,m}$ and $i_{m,s}$ follows from Proposition 1(1). If ${\mathcal{X}}$ is a separable metrizable space, these maps are inclusion maps, and hence they are continuous by definition.

The measurability and the continuity of the vertical mappings $S_{*}(i_{p,m})$ and $S_{*}{(i_{m,s})}$ follow from the measurability and the continuity of the maps $i_{p,m}$ , $i_{m,s}$, respectively, and Proposition 3(2). The measurability and the continuity of the vertical mappings $P_{*}(i_{p,m})$, $S_{*}{(i_{m,s})}$ and $M_{*}(i_{p,m})$, $M_{*}(i_{m,s})$ follow from the corresponding assertion concerning the functor $P_{*}$, the measurability and the continuity for the horizontal mappings, respectively.

3. The commutativity of the diagram is obvious. This completes the proof of Proposition 4.

Let us recall the following result due to Giry.

Lemma 1

( Giry1982 ) Let $ev_{P}:{\mathcal{P}}^{2}({\mathcal{X}})\to{\mathcal{P}}({\mathcal{X}})$ be defined by

[TABLE]

for all $\nu_{\mathcal{P}}\in{\mathcal{P}}^{2}({\mathcal{X}})$ and $A\in\Sigma_{\mathcal{X}}$.

(1) The composition $ev_{P}\circ\delta:{\mathcal{P}}({\mathcal{X}})\to{\mathcal{P}}({\mathcal{X}})$ is the identity map.

(2) Assume that $\kappa:{\mathcal{X}}\to{\mathcal{Y}}$ is a measurable mapping. Then we have the following commutative diagrams

[TABLE]

(3) The mapping $ev_{P}$ is a measurable mapping. It is $\tau_{w}$-continuous, if ${\mathcal{X}}$ is a separable metrizable space.

Note that Giry considered the smaller category of Polish spaces but his proof is also valid for the category of separable metrizable spaces.

Theorem 2.1

(1) Probabilistic morphisms $T$ are morphisms in the category of measurable spaces ${\mathcal{X}}$. Furthermore $(\delta,M_{*})$ and $(\delta,P_{*})$ are faithful functors from the category of measurable spaces whose morphisms are probabilistic morphisms to the category of nonnegative finite measure spaces and the category of probability measure spaces respectively, whose morphisms are measurable mappings. If $\overline{T}:{\mathcal{X}}\to{\mathcal{P}}({\mathcal{Y}})$ is a continuous mapping between separable metrizable spaces then $M_{*}(T):{\mathcal{M}}({\mathcal{X}})\to{\mathcal{M}}({\mathcal{Y}})$ and $P_{*}(T):{\mathcal{P}}({\mathcal{X}})\to{\mathcal{P}}({\mathcal{Y}})$ are $\tau_{w}$-continuous.

(2) If $\nu\ll\mu\in{\mathcal{M}}^{*}({\mathcal{X}})$ then $M_{*}(T)(\nu)\ll M_{*}(T)(\mu)$.

Proof

1. The statement of Theorem 2.1(1) concerning the functor $P_{*}$ without mentioning its faithfulness is a consequence of Giry’s theorem stating that the triple $(P_{*},\delta,ev_{P})$ is a monad in the category of measurable spaces whose morphisms are measurable mappings and in the category of Polish spaces whose morphisms are continuous mappings, respectively, see also the remark before Theorem 2.1 and Remark 6 below) (Giry1982, , Theorem 1, p. 70) and his observation that the Kleisli category of the monad $(P_{*},\delta,ev_{P})$ is the category of measurable spaces whose morphisms are probabilistic morphisms. In other words, Giry’s theorem says that Theorem 2.1 is valid in the subcategory of measurable spaces whose morphisms are measurable mappings, $\delta$ is a natural transformation of the functor $Id_{\mathcal{X}}$ to the functor $P_{*}$ on this subcategory and $ev_{P}$ is a natural transformation of the functor $P_{*}^{2}$ to the the functor $P_{*}$ on the same subcategory. Note that the last assertion is equivalent to the statement of Lemma 1(2) (the associativity of $ev_{P}$ follows from the identity (22) below).

Taking into account Giry’s theorem, the statement of Theorem 2.1 concerning the functoriality (without faithfullness) of $P_{*}$ is a consequence of (MacLane1994, , Theorem 1, p. 143) on the structure of the Kleisli category of a monad. Since the proof of (MacLane1994, , Theorem 1, p.143) is only sketched, we shall prove Proposition 5 below, which proves the functoriality of $P_{*}$. Our proof has also a different strategy than the one sketched in (MacLane1994, , p. 143). Since $M_{*}(T)(c\cdot\mu)=c\cdot M_{*}(T)(\mu)$ for any $c\in{\mathbb{R}}_{\geq 0}$ and $\mu\in{\mathcal{P}}({\mathcal{X}})$, the functoriality of $M_{*}(T)$ is a consequence of the functoriality of $P_{*}(T)$, and the measurability of $M_{*}(T)$ is a consequence of measurability of $P_{*}(T)$. We provide furthermore a new categorical proof for the associativity of the composition of the Markov kernels (Proposition 6).

Proposition 5

Let $T_{i}:{\mathcal{X}}_{i}\leadsto{\mathcal{X}}_{i+1}$ be probabilistic morphisms for $i=2,3$. Then we have

[TABLE]

Proof

The proof consists of two steps. In the first step, using (5) we shall prove relations (10) and (11) below. (If we want to construct the Kleisli category of a monad, we take these formulas as defining relations for the functor $P_{*}$ and for the composition rule of morphisms in the Kleisli category.) Let $T$ and $T_{i}:{\mathcal{X}}_{i}\leadsto{\mathcal{X}}_{i+1}$ be probabilistic morphisms for $i=1,2$. Then

[TABLE]

[TABLE]

Note that (10) is a consequence of the following straightforward computation for any $B\in\Sigma_{\mathcal{Y}}$ and any $\mu\in{\mathcal{P}}({\mathcal{X}})$,

[TABLE]

We leave the reader to verify (11), which has been verified by Giry Giry1982 as we mentioned above, (we prefer to consider (11) as a defining relation and hence we did not recall the known composition rule for Markov kernels in this paper).

In the second step we examine the following diagram:

[TABLE]

By (10), (11), and using (6), we have

[TABLE]

Lemma 2

((Giry1982, , p. 71)) We have the following identity

[TABLE]

From Lemma 1 (3) we obtain immediately the following identity

[TABLE]

Plugging (22) and (23) into (21), taking into account (10), we obtain immediately the first identity of (9), which also implies the second one. This completes the proof of Proposition 5.

Proposition 6

Let $T_{i}:{\mathcal{X}}_{i}\leadsto{\mathcal{X}}_{i+1}$ be probabilistic morphisms for $i=1,2,3$. Then we have

[TABLE]

Proof

Proposition 6 follows from known properties of Markov kernels. Here we give another short algebraic (categorical) proof. Straightforward computations using (11) and (10) yield

[TABLE]

[TABLE]

By Proposition 5 the LHS of (25) equals the LHS of (26). This completes the proof of Proposition 6.

Combining Proposition 5 with Propositions 6 and 3, we prove the assertion of Theorem 2.1(1) concerning the functoriality of $(\delta,P_{*})$ and $(\delta,M_{*})$.

Clearly $\delta:{\mathcal{X}}\to{\mathcal{P}}({\mathcal{X}})$ is an injective map. Using the following equality

[TABLE]

for any $x\in{\mathcal{X}}$ we conclude that $P_{*}$ and $M_{*}$ are injective functors. This completes the proof of Theorem 2.1(1).

2. Theorem 2.1(2) has been proved by Morse-Sacksteder (MS1966, , Proposition 5.1). We now present an alternative proof that is a bit shorter. By the (10) and the validity of Theorem 2.1(2) in the case $T$ is a measurable mapping, it suffices to verify that $ev_{P}(\nu)\ll ev_{P}(\mu)$ which is obvious. This completes the proof of Theorem 2.1.

Remark 6

(1) Theorem 2.1 suggests that to study the category $PMeas$ of measurable spaces whose morphisms are probabilistic morphims we can consider its embedding into the category $({\mathcal{P}}({\mathcal{X}}),\Sigma_{w})$ whose morphisms are generated by measurable mappings between ${\mathcal{X}}$ and ${\mathcal{Y}}$, which are the restriction of linear bounded mappings between Banach spaces ${\mathcal{S}}({\mathcal{X}})$ and ${\mathcal{S}}({\mathcal{Y}})$ endowed with the total variation norm.

(2) Lawvere introduced the $\sigma$-algebra $\Sigma_{w}$ on ${\mathcal{P}}({\mathcal{X}})$ and defined the probabilistic map $ev$ in terms of Markov kernels Lawvere1962 . He coined the term of a “probabilistic mapping” and noted that it is equivalent to the notion of a Markov kernel. He also defined $P_{*}(T)$ by a formula equivalent to (5). In Giry1982 Giry considered the category of measurable spaces whose morphisms are measurable mappings and the category of Polish spaces ${\mathcal{X}}$ whose morphisms are continuous mappings. He noticed that the space $({\mathcal{P}}({\mathcal{X}}),\tau_{w})$ is again Polish cf. (GH1989, , Theorem 2.3). He proved Theorem 2.1 for the case that $P$ is a measurable mapping and ${\mathcal{X}}$ is a Polish space. The triple $(P_{*},\delta,ev_{P})$ is now called Giry’s probability monad. Giry noticed that the Kleisli category of the probability monad $(P_{*},\delta,ev_{P})$ is the category of measurable spaces whose morphisms are probabilistic morphisms. We refer the reader to FP2018 , FP2018 for further investigation of probability monads. Panagaden Panangaden1999 , Panangaden2009 extended the Giry theory to the category SRel (stochastic relations) whose objects are measurable spaces and whose morphisms are transition subprobabilities, but he did not use the concept of $\sigma$-algebra on the space of subprobabilities. Recently Fritz extended the category of probabilistic morphisms to the Markov category Fritz2020 , using high-level axioms, following works of Golubtsov, especially Golubtsov2002 , and the work by Cho and Jacobs CJ2019 . Parzygnat extended Fritz’s theory further Parzygnat2020 , in particular, to include quantum probabilities.

(3) Chentsov called the category of Markov kernels the statistical category and their morphisms - Markov morphisms Chentsov1965 ; Chentsov1972 . Chentsov also showed that $S_{*}(T)$ is a linear operator with $||S_{*}(T)||=1$ for any Markov kernel $T$ and that $S_{*}(T)$ sends a probability measure to a probability measure (Chentsov1972, , Lemma 5.9, p.72). Markov morphisms $S_{*}(T)$ have been investigated further in AJLS2015 , AJLS2017 , AJLS2018 .

(4) Historically, Blackwell was the first who considered probabilistic morphisms, which he called stochastic mappings, between parameterized statistical models, which he called statistical experiments Blackwell1953 . LeCam LeCam1964 used the equivalent terminology randomized mapping and Chentsov Chentsov1965 used the equivalent notion of transition measure.

2.3 Statistical models and sufficient probabilistic morphisms

Given a probabilistic morphism $T$ (for instance, a measurable mapping $\kappa$), we shall also use the short notation $T_{*}$ and $\kappa_{*}$ for $M_{*}(T)$ and $M_{*}(\kappa)$, respectively.

Definition 3

A statistical model is a pair $({\mathcal{X}},P_{\mathcal{X}})$ where ${\mathcal{X}}$ is a measurable space and $P_{\mathcal{X}}\subset{\mathcal{P}}({\mathcal{X}})$. The category of statistical models consists of statistical models as its objects whose morphisms $\varphi:({\mathcal{X}},P_{\mathcal{X}})\leadsto({\mathcal{Y}},P_{\mathcal{Y}})$ are probabilistic morphisms $T:{\mathcal{X}}\leadsto{\mathcal{Y}}$ such that $T_{*}(P_{\mathcal{X}})\subset P_{\mathcal{Y}}$. A morphism $T:({\mathcal{X}},P_{\mathcal{X}})\leadsto({\mathcal{X}},P_{\mathcal{X}})$ will be called a unit, if $T_{*}:P_{\mathcal{X}}\to P_{\mathcal{X}}$ is the identity. Two statistical models $({\mathcal{X}},P_{\mathcal{X}})$ and $({\mathcal{Y}},P_{\mathcal{Y}})$ are called equivalent, if there exist morphisms $T_{1}:({\mathcal{X}},P_{\mathcal{X}})\leadsto({\mathcal{Y}},P_{\mathcal{Y}})$ and $T_{2}:({\mathcal{Y}},P_{\mathcal{Y}})\leadsto({\mathcal{X}},P_{\mathcal{X}})$ such that $T_{1}\circ T_{2}$ and $T_{2}\circ T_{1}$ are units. In this case $T_{1}$ and $T_{2}$ will be called equivalences.

Remark 7

(1) The Kleisli category of Giry’s probability monad, also called the Kleisli category of probabilistic morphisms, can be realized as a subcategory of the category of statistical models by assigning each measurable space ${\mathcal{X}}$ the pair $({\mathcal{X}},{\mathcal{P}}({\mathcal{X}}))$.

(2) The notion of statistical models and their morphisms in Definition 3 is almost equivalent to the notion of statistical systems and their morphisms, called statistical operations, introduced by Morse-Sacksteder MS1966 , except that Morse-Sacksteder allowed morphisms that need not to be induced from probabilistic morphisms (they considered conditional probability as defining statistical operations), and almost coincides with the Markov category of family of probability distributions introduced by Chentsov (Chentsov1972, , §6, p. 76), except that Chentsov considered only morphisms between statistical models whose probability sets $P_{\mathcal{X}},P_{\mathcal{Y}}$ are parameterized by the same set $\Theta$.

Among probabilistic morphisms defining equivalences between statistical models, there is an important class of sufficient morphisms, introduced by Morse-Sacksteder, which we reformulate as follows.

Definition 4

(cf. MS1966 ) A morphism $T:({\mathcal{X}},P_{\mathcal{X}})\leadsto({\mathcal{Y}},P_{\mathcal{Y}})$ between statistical models will be called sufficient if there exists a probabilistic morphism $\underline{{\mathbf{p}}}:{\mathcal{Y}}\leadsto{\mathcal{X}}$ such that for all $\mu\in P_{\mathcal{X}}$ and $h\in L({\mathcal{X}})$ we have

[TABLE]

In this case we shall call $T:{\mathcal{X}}\leadsto{\mathcal{Y}}$ a probabilistic morphism sufficient for ${\mathcal{P}}_{\mathcal{X}}$ and we shall call the measurable mapping ${\mathbf{p}}:{\mathcal{Y}}\to{\mathcal{P}}({\mathcal{X}})$ defining the probabilistic morphism $\underline{{\mathbf{p}}}:{\mathcal{Y}}\leadsto{\mathcal{X}}$ a conditional mapping for $T$.

Remark 8

(1) We call ${\mathbf{p}}:{\mathcal{Y}}\to{\mathcal{P}}({\mathcal{X}})$ a conditional mapping because of the interpretation of ${\mathbf{p}}$ as a regular conditional probability in the case $T$ is a measurable mapping, see (28) below.

(2) As in Remark 2 (1), we note that to prove the sufficiency of a probabilistic morphism $T$ w.r.t. $P_{\mathcal{X}}\subset{\mathcal{P}}({\mathcal{X}})$ it suffices to verify (27) for all $\mu\in P_{\mathcal{X}}$ and for all $h=1_{A}$ where $A\in\Sigma_{\mathcal{X}}$.

Example 3

Assume that $\kappa:({\mathcal{X}},P_{\mathcal{X}})\leadsto({\mathcal{Y}},P_{\mathcal{Y}})$ is a sufficient morphism, where $\kappa:{\mathcal{X}}\to{\mathcal{Y}}$ is a statistic. Let ${\mathbf{p}}:{\mathcal{Y}}\to{\mathcal{P}}({\mathcal{X}}),\,y\mapsto{\mathbf{p}}_{y},$ be a conditional mapping for $\kappa$. By (4), $\underline{{\mathbf{p}}}^{*}(1_{A})(y)={\mathbf{p}}_{y}(A)$, and we rewrite (27) as follows

[TABLE]

The RHS of (28) is the conditional measure of $\mu$ applied to $A$ w.r.t. the measurable mapping $\kappa$. The equality (28) implies that this conditional measure is regular and independent of $\mu$. Thus the notion of sufficiency of $\kappa$ for $P_{\mathcal{X}}$ coincides with the classical notion of sufficiency of $\kappa$ for $P_{\mathcal{X}}$, see e.g., (Chentsov1972, , p. 28), (Schervish1997, , Definition 2.8, p. 85). We also note that the equality in (28) is understood as equivalence class in $L^{1}({\mathcal{Y}},\kappa_{*}(\mu))$ and hence every statistic $\kappa^{\prime}$ that coincides with a sufficient statistic $\kappa$ except on a zero $\mu$-measure set, for all $\mu\in P_{\mathcal{X}}$, is also a sufficient statistic for $P_{\mathcal{X}}$.

Example 4

(cf. (Chentsov1972, , Lemma 2.8, p. 28)) Assume that $\mu\in{\mathcal{P}}({\mathcal{X}})$ has a regular conditional distribution w.r.t. to a statistic $\kappa:{\mathcal{X}}\to{\mathcal{Y}}$, i.e., there exists a measurable mapping ${\mathbf{p}}:{\mathcal{Y}}\to{\mathcal{P}}({\mathcal{X}}),y\mapsto{\mathbf{p}}_{y},$ such that

[TABLE]

for any $A\in\Sigma_{\mathcal{X}}$ and $y\in{\mathcal{Y}}$. Let $\Theta$ be a parameter set and $P:=\{\nu_{\theta}\in{\mathcal{P}}({\mathcal{X}})|\,\theta\in\Theta\}$ be a family of probability measures dominated by $\mu$. If there exist a function $h:{\mathcal{Y}}\times\Theta\to{\mathbb{R}}$ such that for all $\theta\in\Theta$ and we have

[TABLE]

then $\kappa$ is sufficient for $P$, since for any $\theta\in\Theta$

[TABLE]

does not depend on $\theta$. The condition (30) is the Fisher-Neymann sufficiency condition for a family of dominated measures Neyman1935 .

Let us assume that $T:({\mathcal{X}},P_{\mathcal{X}})\leadsto({\mathcal{Y}},P_{\mathcal{Y}})$ is a sufficient morphism for $\mu\in{\mathcal{P}}({\mathcal{X}})$. Then for any $A\in\Sigma_{\mathcal{X}}$ we have

[TABLE]

Comparing (31) with (5), we conclude that $\mu=\underline{{\mathbf{p}}}_{*}\circ T_{*}(\mu)$. Hence $T_{*}\circ\underline{{\mathbf{p}}}_{*}(T_{*}\mu)=T_{*}\mu$. It follows that $({\mathcal{X}},\mu)$ is equivalent to $({\mathcal{Y}},T_{*}(\mu))$. Hence we get the following

Proposition 7

((MS1966, , Proposition 5.2)) Two statistical models $({\mathcal{X}},P_{\mathcal{X}})$ and $({\mathcal{Y}},P_{\mathcal{Y}})$ are equivalent if there exists a probabilistic morphism $T:{\mathcal{X}}\leadsto{\mathcal{Y}}$ such that $T$ is sufficient for $P_{\mathcal{X}}$ and $T_{*}(P_{\mathcal{X}})=P_{\mathcal{Y}}$.

Remark 9

(1) The converse of Proposition 7 is valid, if $P_{\mathcal{X}}$ is a dominated family (Sacksteder1967, , Theorem 2.1), and it is false without the dominated family condition (Sacksteder1967, , §6).

(2) There are many characterizations of sufficiency of a statistic, see e.g. (Pfanzagl2017, , §2.9, p. 35). In (AJLS2017, , Proposition 5.6, p. 266) Ay-Jost-Lê-Schwachhöfer give a characterization of a sufficient statistic w.r.t. to a regular set $P_{\mathcal{X}}$ of dominated measures that is parameterized by a smooth manifold via the notion of “information loss”.

We complete this section with the following theorem on the category of sufficient morphisms.

Theorem 2.2

(1) A composition of sufficient morphisms between statistical models is a sufficient morphism.

(2) Assume that $T:({\mathcal{X}},P_{\mathcal{X}})\leadsto({\mathcal{Y}},P_{\mathcal{Y}})$ is a sufficient morphism and $T_{*}({\mathcal{P}}_{\mathcal{X}})=P_{\mathcal{Y}}$. Let ${\mathbf{p}}:{\mathcal{Y}}\to{\mathcal{P}}({\mathcal{X}})$ be a conditional mapping for $T$. Then $\underline{{\mathbf{p}}}:{\mathcal{Y}}\leadsto{\mathcal{X}}$ is a sufficient probabilistic morphism for $P_{\mathcal{Y}}$ and $\overline{T}:{\mathcal{X}}\to{\mathcal{P}}({\mathcal{Y}})$ is a conditional mapping for $\underline{{\mathbf{p}}}$.

Proof

1. Assume that $T_{1}:({\mathcal{X}},P_{\mathcal{X}})\leadsto({\mathcal{Y}},P_{\mathcal{Y}})$ and $T_{2}:({\mathcal{Y}},P_{\mathcal{Y}})\leadsto({\mathcal{Z}},P_{\mathcal{Z}})$ are sufficient morphisms with conditional mappings ${\mathbf{p}}_{1}:{\mathcal{Y}}\to{\mathcal{P}}({\mathcal{X}})$ and ${\mathbf{p}}_{2}:{\mathcal{Z}}\to{\mathcal{P}}({\mathcal{Y}})$ for $T_{1}$ and $T_{2}$, respectively. Then we have for any $h\in L({\mathcal{X}})$ and $\mu\in P_{\mathcal{X}}$

[TABLE]

[TABLE]

which proves the assertion (1) of Theorem 2.2.

2. Assume that $T:({\mathcal{X}},P_{\mathcal{X}})\to({\mathcal{Y}},P_{\mathcal{Y}})$ is a sufficient morphism and ${\mathcal{P}}_{\mathcal{Y}}=T_{*}(P_{\mathcal{X}})$. Let ${\mathbf{p}}:{\mathcal{Y}}\to{\mathcal{P}}({\mathcal{X}})$ be a conditional mapping for $T$, i.e., we have for any $h\in L({\mathcal{X}})$ and $\mu\in P_{\mathcal{X}}$

[TABLE]

To prove the assertion (2) of Theorem 2.2 it suffices to establish the following equality

[TABLE]

for any $\mu\in P_{\mathcal{X}}$, $B\in\Sigma_{\mathcal{Y}}$, $A\in\Sigma_{\mathcal{X}}$ and $h\in L({\mathcal{X}})$.

Straightforward computations yield

[TABLE]

On the other hand we have

[TABLE]

Comparing (34) with (35), we obtain (33). This completes the proof of Theorem 2.2.

Remark 10

(1) Theorem 2.2 implies that in the subcategory of statistical models equivalent to a given statistical model, whose morphisms are sufficient morphisms, every object is a terminal object, so we don’t have the notion of a “minimal sufficient probabilistic morphism” like the notion of a minimal sufficient statistic. Note that a minimal sufficient statistic does not always exist (Pfanzagl2017, , §2.6, p. 24).

(2) We would like to point out that Fritz Fritz2020 treats sufficient statistics in abstract Markov category and reproves several classical results concerning sufficient statistics using a categorical approach.

(3) We refer the reader to Le2020 for further applications of probabilistic morphisms to the category of statistical models and their natural geometry.

3 Probabilistic morphisms and Bayes’ formula

In this section we formalize the notion of a parameter space of a probabilistic model in Bayesian statistics via the notion of a Bayesian statistical model (Definition 5, Example 5, Remark 11). We define the notion of posterior distributions (Definition 6) and express the existence of posterior distributions in terms of equivalence of statistical models (Proposition 8). Then we prove our Main Theorem 1.1 which provides a new formula for the posterior distributions. We derive a consequence (Corollary 2) and prove the consistence of our formula (Proposition 14).

3.1 Bayesian statistical models and probabilistic morphisms

Definition 5

A Bayesian statistical model is a quadruple $(\Theta,\mu_{\Theta},{\mathbf{p}},{\mathcal{X}})$, where ${\mathbf{p}}:\Theta\to{\mathcal{P}}({\mathcal{X}})$ is a measurable mapping and $\mu_{\Theta}\in{\mathcal{P}}(\Theta)$, called a prior measure.

Rephrasing, a Bayesian statistical model is defined by a probabilistic morphism $\underline{{\mathbf{p}}}$ between statistical models $(\Theta,\{\mu_{\Theta}\})$ and $({\mathcal{X}},{\mathcal{P}}({\mathcal{X}}))$.

Recall that, in our paper, if ${\mathcal{X}}$ is a topological space then $\Sigma_{\mathcal{X}}$ is assumed to be the Borel $\sigma$-algebra ${\mathcal{B}}({\mathcal{X}})$ and hence ${\mathcal{P}}({\mathcal{X}})$ consists of Borel measures, if not otherwise specified.

Example 5

(1) Let ${\mathcal{S}}({\mathcal{X}})_{TV}$ denote the Banach space ${\mathcal{S}}({\mathcal{X}})$ endowed with the total variation norm and $d_{H}$ denote the Hellinger distance on ${\mathcal{P}}({\mathcal{X}})$, so the topology on ${\mathcal{P}}({\mathcal{X}})$ induced by $d_{H}$ coincides with the topology on ${\mathcal{P}}({\mathcal{X}})$ induced from the natural inclusion of $i:{\mathcal{P}}({\mathcal{X}})\to{\mathcal{S}}({\mathcal{X}})_{TV}$. Assume that $(M,\mu_{M})$ is a smooth finite dimensional manifold endowed with a volume element $\mu_{M}$ and ${\mathbf{p}}:M\to({\mathcal{P}}({\mathcal{X}}),d_{H})$ is a continuous map, for instance, if ${\mathbf{p}}$ is a smooth map, i.e., the composition $i\circ{\mathbf{p}}:M\to{\mathcal{S}}({\mathcal{X}})_{TV}$ is a smooth map, see e.g., (AJLS2017, , p. 168), (Bogachev2010, , p. 380). We shall show that $(M,\mu_{M},{\mathbf{p}},{\mathcal{X}})$ is a Bayesian statistical model, i.e., we shall show that the map ${\mathbf{p}}:(M,{\mathcal{B}}(M))\to({\mathcal{P}}({\mathcal{X}}),\Sigma_{w})$ is measurable. Denote by $\tau_{\rho}$ the smallest topology on ${\mathcal{S}}({\mathcal{X}})$ such that for all $f\in{\mathcal{F}}_{s}({\mathcal{X}})$ the map $I_{f}:{\mathcal{S}}({\mathcal{X}})\to{\mathbb{R}}$ is continuous. Clearly the topology $\tau_{\rho}$ is weaker than the strong topology generated by the total variation norm. Let $\tau_{\rho}$ also denote the induced topology on ${\mathcal{P}}({\mathcal{X}})$. It follows that the map ${\mathbf{p}}:M\to{\mathcal{P}}({\mathcal{X}})$ is $\tau_{\rho}$-continuous. Hence the map ${\mathbf{p}}:(M,{\mathcal{B}}(M))\to({\mathcal{P}}({\mathcal{X}}),{\mathcal{B}}(\tau_{\rho}))$ is measurable. It is not hard to check that $\Sigma_{w}\subset{\mathcal{B}}(\tau_{\rho})$, and moreover ${\mathcal{B}}(\tau_{w})\subset{\mathcal{B}}(\tau_{\rho})$ with the equality ${\mathcal{B}}(\tau_{w})={\mathcal{B}}(\tau_{\rho})$ if and only if ${\mathcal{X}}$ is countable GH1989 . We conclude that ${\mathbf{p}}:(M,{\mathcal{B}}(M))\to({\mathcal{P}}({\mathcal{X}}),\Sigma_{w})$ is a measurable mapping.

(2) For any $\mu_{\mathcal{P}}\in{\mathcal{P}}^{2}({\mathcal{X}})$ the quadruple $({\mathcal{P}}({\mathcal{X}}),\mu_{\mathcal{P}},Id_{\mathcal{P}},{\mathcal{X}})$ is a Bayesian statistical model.

(3) Let $(\Theta,\mu_{\Theta},{\mathbf{p}},{\Omega}_{\infty})$ be a Bayesian statistical model, where ${\Omega}_{\infty}:=\{{\omega}_{1},\cdots,{\omega}_{\infty}\}$. Then the probabilistic morphism $\underline{{\mathbf{p}}}:\Theta\leadsto{\Omega}_{\infty}$ will be called a random partition of $\Theta$. If $\underline{{\mathbf{p}}}$ is a measurable mapping then it is also called a measurable partition. A random partition $\underline{{\mathbf{p}}}:\Theta\leadsto{\Omega}_{\infty}$ will be called hierarchical if $\underline{{\mathbf{p}}}=\underline{{\mathbf{p}}}_{k}\circ\cdots\circ\underline{{\mathbf{p}}}_{1}$, where $\underline{{\mathbf{p}}}_{1}:\Theta\leadsto{\Omega}_{\infty}$, and $\underline{{\mathbf{p}}}_{i}:{\Omega}_{\infty}\leadsto{\Omega}_{\infty}$ for $i\in[2,k]$ are probabilistic morphisms.

Remark 11

(1) We recover the classical (frequentist) definition of a (parameterized) statistical model from Definition 5 by applying the “forgetful functor” from the category of measurable spaces to the category of sets.

(2) Definition 5 encompasses both parametric Bayesian models, which usually consist of conditional density functions $f(x|\theta)$, see e.g. (Berger1993, , p. 4), and nonparametric Bayesian models. It is essentially equivalent to the concept of a Bayesian parameter space defined in (GR2003, , p. 16) but we use the more compact functorial language of probabilistic morphisms.

3.2 Posterior distributions and probabilistic morphisms

Let $(\Theta,\mu_{\Theta},{\mathbf{p}},{\mathcal{X}})$ be a Bayesian statistical model. Then we define the joint distribution $\mu$ on the measurable space $(\Theta\times{\mathcal{X}},\Sigma_{\Theta}\otimes\Sigma_{\mathcal{X}})$ as follows

[TABLE]

where we re-denote ${\mathbf{p}}(\theta)$ by ${\mathbf{p}}_{\theta}$ for interpreting $\theta$ as parameter of the distribution. In other words, denoting by $\Gamma_{{\mathbf{p}}}:=(Id_{\Theta},{\mathbf{p}}):\Theta\leadsto(\Theta\times{\mathcal{X}})$ the graph of the probabilistic morphism ${\mathbf{p}}$, then $\mu=(\Gamma_{{\mathbf{p}}})_{*}\mu_{\Theta}.$ As in (2), $\Pi_{\Theta}$ and $\Pi_{\mathcal{X}}$ denote the measurable projections $(\Theta\times{\mathcal{X}},\Sigma_{\Theta}\otimes\Sigma_{\mathcal{X}})\to(\Theta,\Sigma_{\Theta})$ and $(\Theta\times{\mathcal{X}},\Sigma_{\Theta}\otimes\Sigma_{\mathcal{X}})\to({\mathcal{X}},\Sigma_{\mathcal{X}})$ respectively. Using $\Pi_{\mathcal{X}}$ we express the marginal probability measure $\mu_{\mathcal{X}}$ on ${\mathcal{X}}$ as follows: $\mu_{\mathcal{X}}=(\Pi_{\mathcal{X}})_{*}(\mu)$. In other words we have $\mu_{\mathcal{X}}={\mathbf{p}}_{*}(\mu_{\Theta})$. Clearly $\mu_{\Theta}=(\Pi_{\Theta})_{*}(\mu)$. We note that $\mu_{{\mathcal{X}}|\Theta}(A|\theta):={\mathbf{p}}_{\theta}(A)$ is the conditional probability of the measure $\mu$ on $(\Theta\times{\mathcal{X}},\Sigma_{\Theta}\otimes\Sigma_{\mathcal{X}})$ w.r.t. the projection $\Pi_{\Theta}$, since we have the following equality

[TABLE]

cf. (2).

Formula (37) motivates the following definition.

Definition 6

A family of probabilistic measures $\mu_{\Theta|{\mathcal{X}}}(\cdot|x)\in{\mathcal{P}}(\Theta)$, $x\in{\mathcal{X}}$, is called a family of posterior distributions of $\mu_{\Theta}$ after seeing the data $x$ if for all $B\in\Sigma_{\Theta}$ we have

[TABLE]

where (38) should be understood as an equivalence class of functions in $L^{1}({\mathcal{X}},\mu_{\mathcal{X}})$.

Remark 12

(1) Definition 6 coincides with the definition of a posterior distribution in classical Bayesian statistics see e.g. (Berger1993, , §4.2.1, p. 126), (Schervish1997, , p. 16). Note that in the both definitions of the mentioned books the authors did not explicitly require that $\mu_{\Theta|{\mathcal{X}}}(\cdot|x)$ must be a $\sigma$-additive function on $\Sigma_{\Theta}$. The last requirement is trivially satisfied when one considers only Bayesian statistical models of dominated measures, i.e. there exists a measure $\nu_{0}\in{\mathcal{P}}({\mathcal{X}})$ such that for all $\theta\in\Theta$ we have ${\mathbf{p}}(\theta)\ll\nu_{0}$. In other papers and books, e.g., in Ferguson1973 and GV2017 , statisticians also think of posterior distributions as regular conditional distributions, which is equivalent to Definition 6. The existence of posterior distributions is therefore equivalent to the existence of regular conditional distributions.

(2) Alternatively, we can also consider the existence of posterior distributions as the existence of disintegration CP1997 , (Bogachev2007, , p. 380, vol.2), CJ2019 , Parzygnat2020 .

The following Proposition generalizes (Parzygnat2020, , Theorem 2.1), which considers discrete spaces.

Proposition 8

Given a Bayesian statistical model $(\Theta,\mu_{\Theta},{\mathbf{p}},{\mathcal{X}})$. A family of posterior distributions $\mu_{\Theta|{\mathcal{X}}}(\cdot|x)$ exists if and only if the statistical models $(\Theta,\{\mu_{\Theta}\})$ and $({\mathcal{X}},\{\mu_{\mathcal{X}}\})$ are equivalent, where $\mu_{\mathcal{X}}$ is the marginal distribution of ${\mathcal{X}}$, i.e., $\mu_{\mathcal{X}}={\mathbf{p}}_{*}(\mu_{\Theta})\in{\mathcal{P}}({\mathcal{X}})$.

Proof

First we assume that a family of posterior distributions $\mu_{\Theta|{\mathcal{X}}}(\cdot|x)$ exists. As before we let $\mu:=(Id_{\Theta},{\mathbf{p}})_{*}\mu_{\Theta}\in{\mathcal{P}}(\Theta\times{\mathcal{X}})$. Then we define a probabilistic morphism $Q:{\mathcal{X}}\leadsto\Theta$ whose generating measurable mapping $\overline{Q}:{\mathcal{X}}\to{\mathcal{P}}(\Theta)$ is defined by

[TABLE]

Let us compute for $A\in\Sigma_{\mathcal{X}}$ and $B\in\Sigma_{\Theta}$

[TABLE]

It follows that $(Q)_{*}\mu_{\mathcal{X}}=(\Pi_{\Theta})_{*}(\mu)=\mu_{\Theta}$. Hence the statistical models $(\Theta,\mu_{\Theta})$ and $({\mathcal{X}},\mu_{\mathcal{X}})$ are equivalent.

Now assume that the statistical models $(\Theta,\mu_{\Theta})$ and $({\mathcal{X}},\mu_{\mathcal{X}})$ are equivalent. Then there exists a probabilistic morphism $Q:{\mathcal{X}}\leadsto\Theta$ such that $Q_{*}(\mu_{\mathcal{X}})=\mu_{\Theta}$. It follows that ${\mathbf{p}}\circ Q_{*}(\mu_{\mathcal{X}})=\mu_{\mathcal{X}}$. Then we have

[TABLE]

It follows from (41) and (37) that $\mu_{\Theta|{\mathcal{X}}}(B|x):=\overline{Q}(x)(B),\,x\in{\mathcal{X}},$ is the required family of posterior distributions. This completes the proof of Proposition 8.

Remark 13

One knows several theorems on the existence of regular conditional distributions, see e.g. CP1997 , (Bogachev2007, , Chapter 10, vol. 3) for surveys. To our knowledge an explicit formula for posterior distributions is given only in the classical Bayes formula and in a few other exceptional cases. The difficulty in deriving an explicit formula for posterior distributions is caused by three facts. Firstly, the conditional probability is expressed in terms of the Radon-Nikodym derivative, whose existence is known but an explicit formula for the Radon-Nikodym derivative is given only in few exceptional cases, see JLT2020 . Secondly, if we have an explicit formula for $\mu_{\Theta|{\mathcal{X}}}(B|x)$ satisfying (38) we need to show that for $\mu_{\mathcal{X}}$-almost all $x\in{\mathcal{X}}$, $\mu_{\Theta|{\mathcal{X}}}(B|x)$ defines a $\sigma$-additive function on $\Sigma_{\Theta}$. Finally to make a posterior distribution formula suitable for Baeysian inference we need to show that the singular null set that appears in the formula remains a null set in the updated marginal measure.

3.3 Proof of Theorem 1.1

Our proof of Theorem 1.1 shall be divided in two steps. In the first step we shall prove Theorem 1.1 for $k=1$. In the second step we shall prove Theorem 1.1 for any $k\in{\mathbb{N}}^{+}$.

Step 1. In this step, assuming $k=1$, we shall establish the existence of a null set $S\subset{\mathcal{X}}$ that satisfies the condition of Theorem 1.1. This will be done by combining Proposition 9, Corollary 1 below with the existence of a countable algebra generating the Borel $\sigma$-algebra of a Souslin space $\Theta$ and a certain property of $\Theta$ (Proposition 13 below). In what follows we use shorthand notations $\nu_{1}$ for the measure $\mu_{\mathcal{X}}=(\Pi_{\mathcal{X}})_{*}\mu$ and $\nu_{2}$ for $(\Pi_{\mathcal{X}})_{*}(1_{B\times{\mathcal{X}}}\mu)$. Clearly $\nu_{2}={\mathbf{p}}_{*}(1_{B}\mu_{\Theta})\in{\mathcal{M}}({\mathcal{X}})$. By Theorem 2.1(2) $\nu_{2}\ll\nu_{1}$. We recall that

[TABLE]

[TABLE]

For any $x\in{\mathcal{X}}$ we set

[TABLE]

where we set $\overline{D}_{\nu_{1}}\nu_{2}(x)=\underline{D}_{\nu_{1}}\nu_{2}(x)=+\infty$ if $\nu_{1}(D_{r}(x))=0$ for some $r>0$.

Furthermore if $\overline{D}_{\nu_{1}}\nu_{2}(x)=\underline{D}_{\nu_{1}}\nu_{2}(x)$ then we set

[TABLE]

which is called the derivative of $\nu_{2}$ with respect to $\nu_{1}$ at $x$.

Proposition 9

There is a measurable subset $S_{B}\subset{\mathcal{X}}$ of zero $\nu_{1}$-measure such that for any $x\in{\mathcal{X}}\setminus S_{B}$ the function $D_{\nu_{1}}\nu_{2}$ is well-defined. Setting $\tilde{D}_{\nu_{1}}\nu_{2}(x):=0$ for $x\in S_{B}$ and $\tilde{D}_{\nu_{1}}{\nu_{2}}(x)=D_{\nu_{1}}\nu_{2}(x)$ for $x\in{\mathcal{X}}\setminus S_{B}$. Then the function $\tilde{D}_{\nu_{1}}{\nu_{2}}:{\mathcal{X}}\to{\mathbb{R}}$ is measurable and serves as the Radon-Nikodym density of the measure $\nu_{2}$ with respect to $\nu_{1}$.

Proof

We note that the inclusion map ${\mathcal{X}}\to(M^{n},g)$ is continuous and hence measurable. Hence the measures $i_{*}(\nu_{1})$ and $i_{*}(\nu_{2})$ are Borel measures on $(M^{n},g)$ and $i_{*}(\nu_{2})\ll i_{*}(\nu_{1})$, since $\nu_{2}\ll\nu_{1}$. Now Proposition 9 follows immediately from Theorem 1.2 in JLT2020 .

Proposition 9 implies that for each $B\in\Sigma_{\mathcal{X}}$ there exists a measurable subset $S_{B}\subset{\mathcal{X}}$ of zero $\mu_{\mathcal{X}}$-measure such that the RHS of (1) is well defined for all $x\in{\mathcal{X}}\setminus S_{B}$, and moreover it coincides with the RHS of (38) as functions in $L^{1}({\mathcal{X}},\mu_{\mathcal{X}})$. We express this fact in the following equivalent statement.

Corollary 1

Let $B\in\Sigma_{\Theta}$. For $x\in({\mathcal{X}}\setminus S_{B})$ let $p(B,x):=\mu_{\Theta|{\mathcal{X}}}(B|x)$ that is defined in (1) and for $x\in S_{B}$ we let $p(B,x):=0$. Then the function $p(B,x):{\mathcal{X}}\to{\mathbb{R}}$ is measurable. Furthermore, for any $A\in\Sigma_{\mathcal{X}}$ we have

[TABLE]

The strategy of the remainder of the proof of the first assertion of Theorem 1.1 for $k=1$ (Step 1) is as follows. By (Bogachev2007, , Corollary 6.7.5, p. 25, vol.2), there exists a countable algebra ${\mathcal{U}}_{\Theta}$ generating ${\mathcal{B}}(\Theta)$. In Lemma 3 we shall show the existence of a subset $S_{0}\subset{\mathcal{X}}$ of zero $\mu_{\mathcal{X}}$-measure such that for any ${x}\in({\mathcal{X}}\setminus S_{0})$ and any $B\in{\mathcal{U}}_{\Theta}$ the RHS of (1) is well-defined and $\mu_{\Theta|{\mathcal{X}}}(\cdot|{x})=p(B,{\bf x})$, moreover $\mu_{\Theta|{\mathcal{X}}}(\cdot|{x})$ is an additive function on ${\mathcal{U}}_{\Theta}$. Then in Proposition 11 we shall apply Proposition 12 below to show that there is a subset $S_{1}\subset({\mathcal{X}}\setminus S_{0})$ of zero $\mu_{\mathcal{X}}$-measure such that for all ${x}\in{\mathcal{X}}\setminus(S_{0}\cup S_{1})$ the function $p(B,{x})=\mu_{\Theta|{\mathcal{X}}}(\cdot|{x})$ is $\sigma$-additive on ${\mathcal{U}}_{\Theta}$, and hence it defines a probability measure on $\Theta$, since $p(\Theta,{x})=1$ for all ${\bf x}\in({\mathcal{X}}\setminus(S_{0}\cup S_{1}))$. Since on $({\mathcal{X}}\setminus(S_{0}\cup S_{1}))$ the function $p(B,{x})$ coincides with $\mu_{\Theta|{\mathcal{X}}}(B,{x})$ this shall complete the proof the first assertion of Theorem 1.1 for $k=1$.

We recall that a family ${\mathcal{K}}$ of subsets of a set $X$ is called a compact class, if for any sequence $K_{n}$ of its elements with $\cup_{n=1}^{\infty}K_{n}=\emptyset$, there exists $N$ such that $\cup_{i=1}^{N}K_{n}=\emptyset$ (Bogachev2007, , Definition 1.4.1, p. 13, vol.1).

Proposition 10

(Bogachev2007*, *, Theorem 1.4.3, p. 13, vol. 1)** Let $\mu$ be a nonnegative additive set function on an algebra ${\mathcal{A}}$. Suppose that there exists a compact class ${\mathcal{K}}$ approximating $\mu$ in the following sense: for every $A\in{\mathcal{A}}$ and every ${\varepsilon}>0$, there exist $K_{\varepsilon}\in{\mathcal{K}}$ and $A_{\varepsilon}\in{\mathcal{A}}$ such that $A_{\varepsilon}\subset K_{\varepsilon}\subset A$ and $\mu(A\setminus A_{\varepsilon})<{\varepsilon}$. Then $\mu$ is countably additive. In particular this is true if the compact class ${\mathcal{K}}$ is contained in ${\mathcal{A}}$ and for any $A\in{\mathcal{A}}$ one has the equality

[TABLE]

Continuation of Step 1. By Proposition 13 any family ${\mathcal{K}}$ of compact sets in $\Theta$ is a compact class. Propositions 13 and 12 imply that there exists a countable algebra ${\mathcal{U}}_{\Theta}$ generating ${\mathcal{B}}(\Theta)$ such that ${\mathcal{U}}_{\Theta}$ contains a countable union ${\mathcal{K}}_{\Theta}$ of metrizable compact sets on which the measure $\mu_{\Theta}$ is concentrated, and for every $B\in{\mathcal{B}}(\Theta)$ and every ${\varepsilon}>0$ there exists a metrizable compact set $K_{\varepsilon}$ such that $\mu(B\setminus K_{\varepsilon})<{\varepsilon}$. Thus the condition of (Bogachev2007, , Theorem 10.4.5 (ii), p. 359, vol.2) is satisfied for ${\mathcal{A}}={\mathcal{U}}_{\Theta}$.

Since $\Theta$ is a Souslin space, by (Bogachev2007, , Corollary 6.7.5, p. 25, vol.2), there exists a countable algebra ${\mathcal{U}}_{\Theta}$ generating ${\mathcal{B}}(\Theta)$. Let ${\mathcal{U}}_{\Theta}$ consist of countably many sets $B_{n}\in\Sigma_{\Theta}$.

Lemma 3

There exists a measurable subset $S_{0}\subset{\mathcal{X}}$ of zero $\mu_{\mathcal{X}}$-measure such that for all $x\in({\mathcal{X}}\setminus S_{0})$ the function $\mu_{\Theta|{\mathcal{X}}}(\cdot|x)$ is additive on ${\mathcal{U}}_{\Theta}$, moreover $\mu_{\Theta|{\mathcal{X}}}(\cdot|x)=p(B,x)$.

Proof

Let $S_{0}=\cup_{i}S_{B_{i}}$. Then $S_{0}$ is measurable and it has zero measure set. By Corollary 1, $\mu_{\Theta|{\mathcal{X}}}(\cdot|x)=p(B,x)$ for any $x\in({\mathcal{X}}\setminus S_{0})$. Clearly the RHS of (1) is additive for on ${\mathcal{U}}_{\Theta}$ for any $x\in({\mathcal{X}}\setminus S_{0})$. This completes the proof of Lemma 3

Now for all $x\in{\mathcal{X}}$ we define a set function $\tilde{\mu}_{\Theta|{\mathcal{X}}}(\cdot|x)$ on ${\mathcal{U}}_{\Theta}$ by setting

[TABLE]

By Corollary 1 and Lemma 3, for any $B\in{\mathcal{U}}_{\Theta}$ the function $\tilde{\mu}_{\Theta|{\mathcal{X}}}(B|\cdot):{\mathcal{X}}\to{\mathbb{R}}$ is measurable.

Proposition 11

There exists a measurable subset $S_{1}\subset{\mathcal{X}}\setminus S_{0}$ of zero $\mu_{\mathcal{X}}$-measure such that for any $x\in({\mathcal{X}}\setminus(S_{0}\cup S_{1}))$ the set function $\tilde{\mu}_{\Theta|{\mathcal{X}}}(\cdot|x)$ is $\sigma$-additive on ${\mathcal{U}}_{\Theta}$.

For the proof of Proposition 11 we shall use Propositions 12, 13 below.

We recall that a family ${\mathcal{K}}$ of subsets of a set $X$ is called a compact class, if for any sequence $K_{n}$ of its elements with $\cup_{n=1}^{\infty}K_{n}=\emptyset$, there exists $N$ such that $\cup_{i=1}^{N}K_{n}=\emptyset$. In particular, an arbitrary family of compact sets in a topological space is a compact class (Bogachev2007, , p. 13, vol.1).

Proposition 12

(Bogachev2007*, *, Theorem 1.4.3, p. 13, vol. 1)** Let $\mu$ be a nonnegative additive set function on an algebra ${\mathcal{A}}$. Suppose that there exists a compact class ${\mathcal{K}}$ approximating $\mu$ in the following sense: for every $A\in{\mathcal{A}}$ and every ${\varepsilon}>0$, there exist $K_{\varepsilon}\in{\mathcal{K}}$ and $A_{\varepsilon}\in{\mathcal{A}}$ such that $A_{\varepsilon}\subset K_{\varepsilon}\subset A$ and $\mu(A\setminus A_{\varepsilon})<{\varepsilon}$. Then $\mu$ is countably additive. In particular this is true if the compact class ${\mathcal{K}}$ is contained in ${\mathcal{A}}$ and for any $A\in{\mathcal{A}}$ one has the equality

[TABLE]

Proposition 13

((Bogachev2007, , Theorem 7.4.3, p. 85, vol. 2)) Every Borel measure $\mu$ on a Souslin space $\Theta$ is Radon and concentrated on a countable union of metrizable compact sets. In addition, for every $B\in{\mathcal{B}}(\Theta)$ and every ${\varepsilon}>0$ there exists a metrizable compact set $K_{\varepsilon}$ such that $\mu(B\setminus K_{\varepsilon})<{\varepsilon}$.

Proof (Proof of Proposition 11)

By Proposition 13 there exists a compact class ${\mathcal{C}}$ consisting of compact sets $C_{n,l}$ such that for all $n,l\in{\mathbb{N}}^{+}$ we have

[TABLE]

Lemma 4

There exists a measurable subset $S_{1}\subset{\mathcal{X}}\setminus S_{0}$ of zero $\mu_{\mathcal{X}}$-measure such that for all $n,l\in{\mathbb{N}}^{+}$ and for all $x\in({\mathcal{X}}\setminus(S_{0}\cup S_{1}))$ and we have

[TABLE]

Proof

We define a function $q_{n}:{\mathcal{X}}\to{\mathbb{R}}$ by setting $q_{n}(x)$ equal to the RHS of (47), if $x\in({\mathcal{X}}\setminus S_{0})$ and $q_{n}(x)=0$ otherwise. Then $q_{n}:{\mathcal{X}}\to{\mathbb{R}}$ is measurable, since $\tilde{\mu}_{\Theta|{\mathcal{X}}}(C_{n,l}|\cdot):{\mathcal{X}}\to{\mathbb{R}}$ is measurable. Since $C_{n,l}\subset B_{n}$ for all $l$, we have for all $x\in({\mathcal{X}}\setminus S_{0})$,

[TABLE]

Since $\mu_{\Theta|{\mathcal{X}}}(C_{n,l}|x)\leq q_{n}(x)$ for $x\in{\mathcal{X}}\setminus S_{0}$, and $S_{0}$ is a measurable subset of zero $\mu$-measure, taking into account (44) and (45), we have

[TABLE]

Taking into account (48) and (46), we obtain from (49)

[TABLE]

By (46), the far LHS of (50) is equal to the far RHS of (50). Taking into account (48), we conclude that there exists a subset $S_{1}\subset{\mathcal{X}}\setminus S_{0}$ of zero $\mu_{\mathcal{X}}$-measure such that $q_{n}(x)=\mu_{\Theta|{\mathcal{X}}}(x)$ for all $x\in{\mathcal{X}}\setminus(S_{0}\cup S_{1})$. Since both the functions $q_{n}$ and $\mu_{\Theta|{\mathcal{X}}}$ are measurable, the subset $S_{1}$ is measurable. This proves Lemma 4.

Lemma 4 implies that for all $x\in{\mathcal{X}}\setminus(S_{0}\cup S_{1})$ the additive function $\tilde{\mu}_{\Theta|{\mathcal{X}}}(\cdot|x)$ on the algebra ${\mathcal{U}}_{\Theta}$ has the property that the compact class ${\mathcal{K}}_{\Theta}$ approximates $p(\cdot,x)$ on ${\mathcal{U}}_{\Theta}$. Combining with Proposition 12, this completes the proof of Proposition 11.

Since $\tilde{\mu}_{\Theta|{\mathcal{X}}}(\cdot|x)$ is $\sigma$-additive on ${\mathcal{U}}_{\Theta}$, it extends to a measure on $\Theta$. Since $\tilde{\mu}_{\Theta|{\mathcal{X}}}(\Theta,x)=1$ for any $x\in{\mathcal{X}}$ this completes the proof of Theorem 1.1 for the case $k=1$.

Corollary 2

Assume the conditions of Theorem 1.1. Given a point $x_{0}\in{\mathcal{X}}\setminus S$ assume that ${\mathbf{p}}_{\theta}(x_{0})=0$ for all $\theta\in\Theta$. If the condition for differentiation w.r.t. $r$ at [math] under the integral $\int_{C}{\mathbf{p}}_{\theta}(D_{r}(x_{0}))d\mu_{\Theta}$ holds for $C\in{\mathcal{U}}_{\Theta}\cup\{\Theta\}$, then for any $B\in{\mathcal{U}}_{\Theta}$ we have

[TABLE]

if the dominator in the RHS of (51) does not vanish.

Remark 14

Assume that $\{{\mathbf{p}}_{\theta}|\>\theta\in\Theta\}$ is a family of measures dominated by a measure $\nu\in{\mathcal{P}}({\mathcal{X}})$, i.e. for each $\theta\in\Theta$ there exists $f_{{\mathcal{X}}|\Theta}(\cdot|\theta)\in L^{1}({\mathcal{X}},\nu)$ such that ${\mathbf{p}}_{\theta}=f_{{\mathcal{X}}|\Theta}(\cdot|\theta)\nu$. Then

[TABLE]

where, by Bayes’ formula we have (Schervish1997, , Theorem 1.31, p.16)

[TABLE]

for $\mu_{\mathcal{X}}$-a.e. $x\in{\mathcal{X}}$. We regard both (1) and (53) as recipes for computing the posterior distribution $\mu(B|x)$ under different assumptions.

Remark 15

Assume the conditions of Theorem 1.1. Given $x_{0}\in{\mathcal{X}}\setminus S$, assuming that ${\mathbf{p}}_{\theta}(x_{0})=0$ for all $\theta\in\Theta$, a sufficient condition for differentiation under the integral sign for any $C\in{\mathcal{U}}_{\Theta}\cup\Theta$

[TABLE]

is the differentiability in $r$ of the function ${\mathbf{p}}_{\theta}(D_{r}(x_{0}))$ of the variable $r$ in a neighborhood of $0\in[0,1]$ and the existence of a $\mu_{\Theta}$-measurable function $F(\theta,x_{0})$ of the variable $\theta$ such that $|\frac{d}{dr}{\mathbf{p}}_{\theta}(D_{r}(x_{0}))|\leq F(\theta,x_{0})$ in this neighborhood, see e.g. (Jost2005, , Theorem 16.11, p. 213), which is also valid for an arbitrary measurable space $\Theta$.

Step 2 Completion of the proof of Theorem 1.1. Let $\mu_{k}$ denote the joint distribution on $\Theta\times{\mathcal{X}}^{k}$ defined by the Bayesian statistical model $(\Theta,\mu_{\Theta},{\mathbf{p}}^{k},{\mathcal{X}}^{k})$. Given $B\in\Sigma_{\Theta}$, we denote by $\nu_{1}^{k}$ the marginal probability measure $\mu_{{\mathcal{X}}^{k}}=(\Pi_{{\mathcal{X}}^{k}})_{*}(\mu_{k})$ and by $\nu_{2}^{k}$ the measure $(\Pi_{{\mathcal{X}}^{k}})_{*}(1_{B\times{\mathcal{X}}^{k}}\mu_{k})$. Since $\nu_{1}^{k}=({\mathbf{p}}^{k})_{*}\mu_{\Theta}$ and $\nu_{2}^{k}=({\mathbf{p}}^{k})_{*}(1_{B}\mu_{\Theta})$ we obtain

[TABLE]

From (55) it follows that the RHS of (1) exists and equals to the LHS of (1). This completes Theorem 1.1.

Proposition 14

For any ${\bf x}=(x_{1},\cdots,x_{k})\in({\mathcal{X}}\setminus S)^{k}$ the set $S$ is also of zero marginal measure $\mu^{{\bf x}}_{\mathcal{X}}$ w.r.t. the updated posterior measure $\mu_{\Theta|{\mathcal{X}}}(\cdot|{\bf x})$

Proof

Let us compute the updated marginal measure of $S$

[TABLE]

since for all $r$ we have

[TABLE]

This completes the proof of Proposition 14.

Proposition 14 implies that our formula for posterior distributions is consistent for Bayes inference see Remark 12(2) and (GV2017, , p.6). We postpone a discussion on posterior consistency in another paper.

Remark 16

We can slightly generalize Theorem 1.1 to the case of a finite dimensional complete Riemannian manifold with a countable number of connected components by setting the distance between two points from different connected components to be $\infty$.

Example 6

Assume that ${\mathcal{X}}\subset(M,g)$ is a closed (or open) subset of $(M,g)$. Then ${\mathcal{X}}$ is a Polish space and ${\mathcal{P}}({\mathcal{X}})$ is a Polish space and hence a Souslin space. Theorem 1.1 implies that for any (prior) measure $\mu_{\Theta}\in{\mathcal{P}}^{2}({\mathcal{X}})$ our formula (1) gives a posterior distribution of $\mu_{\Theta}$ on $\Theta:={\mathcal{P}}({\mathcal{X}})$ after seeing data ${\bf x}\in{\mathcal{X}}^{k}$ for any $k\in{\mathbb{N}}^{+}$. If $\dim{\mathcal{X}}\geq 1$ then ${\mathcal{P}}({\mathcal{X}})$ is not a family of dominated measures and therefore we cannot apply Bayes’ formula for computing the posterior distribution of $\mu_{\Theta}$ after seeing data ${\bf x}$, unless we have a constraint on $\mu_{\Theta}$, e.g., $\mu_{\Theta}$ is a Dirichlet measure, see (GV2017, , Theorem 4.6, p.62) for a short proof of computing the posterior distribution of Dirichlet measures.

4 Dirichlet measures revisited

In this section, first we discuss some functorial methods of generating probability measures on ${\mathcal{P}}({\mathcal{X}})$. Then, by using the functorial language of probabilistic morphisms, we revisit Dirichlet distributions, which are Dirichlet measures on ${\mathcal{P}}({\mathcal{X}})$ where the ${\mathcal{X}}$ are finite sample spaces, see Definition 8 and the remark thereafter. Finally we give a new proof of the existence of Dirichlet measures over any measurable space by using a functorial property of the Dirichlet map constructed by Sethuraman (Theorem 4.1).

4.1 Probability measures on ${\mathcal{P}}({\mathcal{X}})$

There are several known techniques for construction of random measures, i.e., measures on ${\mathcal{P}}({\mathcal{X}})$, see (GV2017, , Chapter 3) for an extensive account. In this section we shall use probabilistic morphisms for construction of a probability measure on ${\mathcal{P}}({\mathcal{X}})$. The most natural way is to look at a “simpler” measurable space ${\mathcal{X}}_{s}$ and construct a probabilistic morphism $T:{\mathcal{X}}_{s}\leadsto{\mathcal{X}}$ together with a measure $\mu\in{\mathcal{P}}({\mathcal{X}}_{s})$, a measure $\mu_{\mathcal{P}}\in{\mathcal{P}}^{2}({\mathcal{X}}_{s})$, and examine if the measures $P_{*}(\overline{T})(\mu),\,ev_{P}\circ P_{*}^{2}(\overline{T})(\mu_{\mathcal{P}}),\,P_{*}^{2}(T)(\mu_{\mathcal{P}})\in{\mathcal{P}}^{2}({\mathcal{X}})$ satisfy our requirement.

[TABLE]

By Lemma 1 (3), $ev_{P}\circ P_{*}^{2}(\overline{T})(\mu_{\mathcal{P}})=P_{*}(\overline{T})(ev_{P}(\mu_{\mathcal{P}}))$, hence we need only to look at $P_{*}(\overline{T})(\mu)$ and $P_{*}^{2}(T)(\mu_{\mathcal{P}})$. Both these constructions are utilized in the proof of Theorem 4.1, where we give an alternative proof of the Sethuraman theorem.

Let us recall that the $\sigma$-algebra on ${\mathcal{P}}({\mathcal{X}})$ is generated by the subsets $\langle A,B^{*}\rangle_{P}:=\langle A,B^{*}\rangle\cap{\mathcal{P}}({\mathcal{X}})$ where $A\in\Sigma_{\mathcal{X}}$ and $B^{*}\in{\mathcal{B}}([0,1])$ (see the proof of Prop 1). Hence we obtain the following easy Lemma whose proof is omitted.

Lemma 5

Any $\mu_{\mathcal{P}}\in{\mathcal{P}}^{2}({\mathcal{X}})$ is determined uniquely by its value on the measurable subsets $\langle A_{1},B_{1}^{*}\rangle_{P}\cap\cdots\cap\langle A_{n},B_{n}^{*}\rangle_{P}$ where $A_{i}\in\Sigma_{\mathcal{X}}$, $B_{i}^{*}\in{\mathcal{B}}([0,1])$ and $n\in{\mathbb{N}}$.

Note that the collection $\{(A_{1}\times B_{1}^{*})\times\cdots\times(A_{n}\times B_{n}^{*})\}$ of measurable subsets generates the $\sigma$-algebra in ${\mathcal{X}}_{univ}:=(({\mathcal{X}}\times[0,1])^{\infty},\Sigma_{{\mathcal{X}}_{univ}}:=(\Sigma_{\mathcal{X}}\otimes{\mathcal{B}}([0,1]))^{\infty})$, moreover any measure $\nu\in{\mathcal{P}}({\mathcal{X}}_{univ})$ is determined uniquely by its values on the subsets in this collection. This suggests that we could take ${\mathcal{X}}_{s}:={\mathcal{X}}_{univ}$ to define a prior measure on ${\mathcal{P}}({\mathcal{X}})$ such that the obtained prior measure on ${\mathcal{P}}({\mathcal{X}})$ satisfies Ferguson’s required properties: the support of the prior distribution should be large and the posterior distributions should be manageable analytically. A required probabilistic morphism ${\mathcal{X}}_{univ}\leadsto{\mathcal{X}}$ has been constructed in Sethuraman’s proof of the existence of Dirichlet measures Sethuraman1994 , which we shall revisit at the end of this section.

4.2 Dirichlet distributions

Dirichlet distributions are most commonly used as the prior distribution over finite sample spaces in Bayesian statistics. In this subsection, following (GR2003, , §3.1.1) and Ferguson1973 , we recall the notion of a Dirichlet distribution $Dir(\alpha_{1},\cdots,\alpha_{k})$ on $\Delta_{k}={\mathcal{P}}({\Omega}_{k})$, where $\alpha:=(\alpha_{1},\cdots,\alpha_{k})\in{\mathbb{R}}^{k}_{\geq 0}\setminus\{0\}$ is a parameter of the distribution. Classically, Dirichlet distributions $Dir(\alpha)$ are defined for $\alpha\in{\mathbb{R}}^{k}_{>0}$ but Ferguson’s definition of a Dirichlet distribution (Definition 7) extends naturally to a definition of a Dirichlet measure in general case (Definition 8) when ${\mathcal{X}}$ need not be finite, while the classical definition doesn’t.chlet distribution on ${\Omega}(\alpha)$.

Definition 7

(cf. (GR2003, , Definition 3.1.1, p. 89), (Ferguson1973, , p. 211)) Given $\alpha\in{\mathcal{M}}^{*}({\Omega}_{k})$ let ${\Omega}(\alpha):=\{{\omega}_{j}\in{\Omega}_{k}|\,\alpha({\omega}_{j})\not=0\}$. Let $\pi_{\alpha}:{\Omega}(\alpha)\to{\Omega}_{k}$ denote the natural inclusion. Let $l(\alpha):=\#{\Omega}(\alpha)$. The Dirichlet distribution $Dir(\alpha)\in{\mathcal{P}}^{2}({\Omega}_{k})={\mathcal{P}}(\Delta_{k})$ is the measure $P_{*}^{2}(\pi_{\alpha})Dir(\alpha|_{{\Omega}(\alpha)})$, where $Dir(\alpha|_{{\Omega}(\alpha)})\in{\mathcal{P}}^{2}({\Omega}(\alpha))$ is the classical $(l(\alpha)-1)$-dimensional Dirichlet distribution on $\Delta_{l(\alpha)}$, i.e. the distribution with the following density function with respect to the $(l(\alpha)-1)$-dimensional Lebesgue measure on $\Delta_{l(\alpha)}$:

[TABLE]

where $(x_{i_{1}},\cdots,x_{i_{l(\alpha)}})\in\Delta_{l(\alpha)}$.

We summarize important known properties of Dirichlet distributions, which shall be needed later, in the following

Proposition 15

(1) The map $Dir_{k}:({\mathcal{M}}^{*}({\Omega}_{k}),\tau_{w})\to({\mathcal{P}}^{2}({\Omega}_{k}),\tau_{w}):\alpha\mapsto Dir(\alpha)$ is continuous (GR2003, , 11(a), p. 93).

*(2) If $p\in\Delta_{k}$ is distributed by $Dir(\alpha_{1},\cdots,\alpha_{k})$ then for any partition $A_{1},\cdots,A_{l}$ of ${\Omega}_{k}$ the vector

$(\sum_{{\omega}_{i}\in A_{1}}p_{i},\sum_{{\omega}_{i}\in A_{2}}p_{i},\cdots,\sum_{{\omega}_{i}\in A_{l}}p_{i})$ is distributed by $Dir(\alpha(A_{1}),\cdots,\alpha(A_{l}))$ (GR2003, , p. 90).*

(3) Let $\alpha\in{\mathcal{M}}^{*}({\Omega}_{k})$ and $({\mathcal{P}}({\Omega}_{k}),Dir(\alpha),Id_{\mathcal{P}},{\Omega}_{k})$ be a Bayesian statistical model. Then the posterior distribution of the prior distribution $Dir(\alpha)$ after seeing the data $x\in{\Omega}_{k}$ is $Dir(\alpha+\delta_{x})$ (Ferguson1973, , p. 212), (GR2003, , p. 92).

Remark 17

(1) Let ${\mathcal{X}}=A_{1}\dot{\cup}\cdots\dot{\cup}A_{n}$ be a measurable partition of ${\mathcal{X}}$ into $n$ disjoint measurable subsets. This partition induces a measurable map $\pi:{\mathcal{X}}\to{\Omega}_{n}:=\{A_{1},\cdots,A_{n}\}$, $\pi(x):=A_{i}$ if $x\in A_{i}$. We have $M_{*}(\pi)(\alpha)=(\alpha(A_{1}),\cdots,\alpha(A_{n}))$ for any $\alpha\in{\mathcal{M}}({\mathcal{X}})$. Hence the assertion (2) in Proposition 15 is equivalent to the commutativity of the following diagram for any $n\geq k$ and any surjective mapping $\pi_{nk}:{\Omega}_{n}\to{\Omega}_{k}$

[TABLE]

(2) Let $k<l$ and $\pi_{kl}:{\Omega}_{k}\to{\Omega}_{l}$ is an injection. Let $\pi_{lk}:{\Omega}_{l}\to{\Omega}_{k}$ be a left inverse of $\pi_{kl}$. By the definition of $Dir(\alpha)$ it is not hard to see that the following diagram is commutative

[TABLE]

It follows that for any map $\pi_{kl}:{\Omega}_{k}\to{\Omega}_{l}$ we have

[TABLE]

Hence $Dir_{k}:{\mathcal{M}}^{*}({\Omega}_{k})\to{\mathcal{P}}^{2}({\Omega}_{k})$ is a natural transformation of the functor $M_{*}$ to the functor $P_{*}^{2}$ in the category of finite sample spaces ${\Omega}_{k}$ whose morphisms are (measurable) mappings.

(3) The converse statement of Remark 17(2) is also valid, assuming a normalization condition. Namely for each $n\in{\mathbb{N}}^{+}$ there exists a unique mapping $Dir_{n}:{\mathcal{M}}^{*}({\Omega}_{n})\to{\mathcal{P}}^{2}({\Omega}_{n})$ satisfying the following properties.

(i) Normalization: for $n=2$ we have $Dir_{2}(\alpha)=Dir(\alpha)$.

(ii) Naturality: for any $n,k\geq 1$ and any mapping $\pi_{nk}:{\Omega}_{n}\to{\Omega}_{k}$ the following diagram is commutative

[TABLE]

4.3 Dirichlet measures

Definition 8

(cf. (Ferguson1973, , Definition 1, p. 214)) Let ${\mathcal{X}}$ be a measurable space and $\alpha\in{\mathcal{M}}^{*}({\mathcal{X}})$. An element ${\mathcal{D}}(\alpha)\in{\mathcal{P}}^{2}({\mathcal{X}})$ is called a Dirichlet measure on ${\mathcal{P}}({\mathcal{X}})$ parameterized by $\alpha$, if for all surjective measurable mappings $\pi_{k}:{\mathcal{X}}\to{\Omega}_{k}$ we have

[TABLE]

where $Dir(\alpha_{1},\cdots,\alpha_{k})$ is the Dirichlet distribution with the parameter $(\alpha_{1},\cdots,\alpha_{k})$ on $\Delta_{k}$ defined in Definition 7. If ${\mathcal{D}}(\alpha)$ is defined for all $\alpha\in{\mathcal{M}}^{*}$ we shall call ${\mathcal{D}}:{\mathcal{M}}^{*}({\mathcal{X}})\to{\mathcal{P}}^{2}({\mathcal{X}})$ a Dirichlet map.

Proposition 15 (2) implies that $Dir(\alpha_{1},\cdots,\alpha_{k})$ is a Dirichlet measure on ${\mathcal{P}}({\Omega}_{k})$.

Theorem 4.1

For any measurable space ${\mathcal{X}}$ there exists a measurable mapping ${\mathcal{D}}:{\mathcal{M}}^{*}({\mathcal{X}})\to{\mathcal{P}}^{2}({\mathcal{X}})$ such that ${\mathcal{D}}(\alpha)$ is a Dirichlet measure parameterized by $\alpha$. Moreover the mapping ${\mathcal{D}}$ is a natural transformation of the functor $M_{*}$ to the functor $P_{*}^{2}$ in the category of measurable spaces whose morphisms are measurable mappings.

Proof

We shall show that the Dirichlet mapping ${\mathcal{D}}:{\mathcal{M}}^{*}({\mathcal{X}})\to{\mathcal{P}}^{2}({\mathcal{X}})$ constructed by Sethuraman Sethuraman1994 satisfies the condition in Theorem 4.1(1). Let us recall the construction of ${\mathcal{D}}$. First we define a mapping $T:{\mathcal{M}}^{*}({\mathcal{X}})\to{\mathcal{P}}({\mathcal{X}}_{univ})$ as follows

[TABLE]

Here, $beta$ is the usual beta distribution. Since $\alpha({\mathcal{X}})>0$, the map $T_{1}:{\mathcal{M}}^{*}({\mathcal{X}})\to{\mathcal{P}}^{*}([0,1]),\alpha\mapsto beta(1,\alpha({\mathcal{X}}))$ is a measurable map. Since $\pi_{\mathcal{X}}$ is a measurable map by Proposition 4, the map $T$ is measurable.

Let $\pi_{i}:{\mathcal{X}}_{univ}\to{\mathcal{X}}\times[0,1]$ denote the projection on the $i$-factor of ${\mathcal{X}}_{univ}$. Denote by $j_{1}$ and $j_{2}$ the projections from ${\mathcal{X}}\times[0,1]$ to ${\mathcal{X}}$ and to $[0,1]$, respectively. The we define for $i\in{\mathbb{N}}$ and $n\in{\mathbb{N}}$ the following measurable mappings:

$\bullet$ $\theta_{i}:{\mathcal{X}}_{univ}\to{\mathbb{R}},\,\theta_{i}(x_{univ}):=j_{2}\circ\pi_{i}(x_{univ})$,

$\bullet$ $q_{i}:{\mathcal{X}}_{univ}\to{\mathcal{X}},\,q_{i}(x_{univ}):=j_{1}\circ\pi_{i}(x_{univ})$,

$\bullet$ $p_{1}:{\mathcal{X}}_{univ}\to{\mathbb{R}},\,p_{1}(x_{univ}):=\theta_{1}(x_{univ})$,

$\bullet$ $p_{n}:{\mathcal{X}}_{univ}\to{\mathbb{R}},\,p_{n}(x_{univ}):=\theta_{n}(x_{univ})\Pi_{i=1}^{n-1}(1-\theta_{i}(x_{univ}))$, if $n\geq 2$,

$\bullet$ $p_{univ}:{\mathcal{X}}_{univ}\to{\mathcal{P}}({\mathcal{X}}),\ p_{univ}(x_{univ}):=\sum_{i=1}^{\infty}p_{i}(x_{univ})\cdot\delta_{q_{i}(x_{univ})}$

$\bullet$ ${\mathcal{D}}:{\mathcal{M}}^{*}({\mathcal{X}})\to{\mathcal{P}}^{2}({\mathcal{X}}),\,{\mathcal{D}}(\alpha):=(p_{univ})_{*}(T_{univ}(\alpha))$.

Remark 18

It is not hard to see that for any $x_{univ}\in{\mathcal{X}}_{univ}$ we have $\sum_{i=1}^{\infty}p_{i}(x_{univ})\in[0,1]$. Therefore the image of $p_{univ}({\mathcal{X}}_{univ})\subset{\mathcal{M}}({\mathcal{X}})$. Furthermore it is known it that there exists a measurable subset ${\mathcal{X}}_{univ}^{reg}$ of full $\mu_{univ}$-measure for any $\mu_{univ}=\otimes^{\infty}(\mu_{\mathcal{X}}\times\nu)\in{\mathcal{P}}({\mathcal{X}}_{univ})$, where $\mu_{\mathcal{X}}\in{\mathcal{P}}({\mathcal{X}})$ and $\nu\in{\mathcal{P}}([0,1])$ is dominated by the Lebesgue measure on $[0,1]$, such that the map $p_{univ}({\mathcal{X}}^{reg})\subset{\mathcal{P}}({\mathcal{X}})$, see e.g. (GV2017, , Lemma 3.4, p. 31). Thus the expression $(p_{univ})_{*}(T(\alpha))$ should be understood as $(p_{univ})_{*}\circ r(\alpha_{univ})\in{\mathcal{P}}^{2}({\mathcal{X}})$ where $r:{\mathcal{P}}({\mathcal{X}}_{univ})\to{\mathcal{P}}({\mathcal{X}}_{univ}^{reg})$ is the restriction and $\alpha_{univ}:=T(\alpha)$.

Since the mappings $p_{i}$ are measurable, and $\sum_{i=1}^{\infty}p_{i}(x_{univ})=1$ for $x_{univ}\in{\mathcal{X}}_{univ}^{reg}$, the map ${\mathcal{D}}(\alpha):{\mathcal{M}}^{*}({\mathcal{X}})\to{\mathcal{P}}^{2}_{*}({\mathcal{X}})$ is a measurable map.

Now we shall show that Sethuraman’s Dirichlet map ${\mathcal{D}}$ is a natural transformation of the functor $M_{*}$ to the functor $P_{*}^{2}$.

Let $\kappa:{\mathcal{X}}\to{\mathcal{Y}}$ be a measurable mapping. Denote by $\kappa_{\infty}:{\mathcal{X}}_{univ}\to{\mathcal{Y}}_{univ}$ the induced measurable mapping:

[TABLE]

Clearly we have $\kappa_{\infty}({\mathcal{X}}_{univ}^{reg})\subset{\mathcal{Y}}_{univ}^{reg}$. Let us consider the following diagram

[TABLE]

Assume that $\alpha\in{\mathcal{M}}^{*}({\mathcal{X}})$. Since $\kappa_{*}(\alpha)({\mathcal{Y}})=\alpha({\mathcal{X}})$ we obtain

[TABLE]

Let $\langle A_{i},B_{i}^{*}\rangle_{P}\in\Sigma_{{\mathcal{P}}({\mathcal{Y}})}$. Then

[TABLE]

[TABLE]

Let $\langle A,B^{*}\rangle_{P}\in\Sigma_{{\mathcal{P}}({\mathcal{Y}})}$. Then we have

[TABLE]

Hence

[TABLE]

It follows from (64), (63) and (62) that

[TABLE]

Taking into account Lemma 5, we obtain from (65) the following identity

[TABLE]

Hence we deduce from (60) and (66) the naturality of the transformation ${\mathcal{D}}$ which is expressed in the following identity

[TABLE]

As in Remark 17, we observe that the functoriality of the map ${\mathcal{D}}$ implies that ${\mathcal{D}}(\alpha)$ is a Dirichlet measure, since it is known that the restriction of the map ${\mathcal{D}}$ to the category of finite sample spaces is the Dirichlet map, see e.g. (GV2017, , §3.3.3). (By Remark 17(3) we need only to show that the Sethuraman map defined on $M_{*}({\Omega}_{2})$ is the Dirichlet map). This completes the proof of Theorem 4.1.

Remark 19

1. Let $\Theta$ denote ${\mathcal{P}}({\mathcal{X}})$. Assume that ${\mathcal{D}}:{\mathcal{M}}^{*}({\mathcal{X}})\to{\mathcal{P}}(\Theta)$ is a Dirichlet map. In (Ferguson1973, , Theorem 1, p. 217) Ferguson proved that the posterior (conditional) distribution ${\mathcal{D}}_{\Theta|{\mathcal{X}}}(\cdot|x)$ with prior ${\mathcal{D}}(\alpha)$ is equal to ${\mathcal{D}}(\alpha+\delta_{x})$.

In Ferguson1973 Ferguson made use of Kolmogorov’s consistency theorem. Since ${\mathcal{P}}({\mathbb{R}},\Sigma_{w})$ is not a measurable subset of $[0,1]^{{\mathcal{B}}({\mathbb{R}})}$, see e.g., (GR2003, , p. 64), the Kolmogorov theorem does not apply directly and we need a more refined technique (GV2017, , Theorem 3.12, p.28). Ferguson also suggested a second proof of the existence of a Dirichlet map, which is close to Sethuraman’s proof Sethuraman1994 .

We have shown the benefit of the functorial language of probabilistic morphisms in this paper. In particular, by using the concise functorial language of probabilistic morphisms, we don’t need to use abstract generators of random variables.

Acknowledgement

The authors would like to thank Tobias Fritz for discussions on probability monads, Markov category and many useful comments on an early version of this paper, Lorenz Schwachhöfer, Juan Pablo Vigneaux for beneficial comments on different versions of this paper, and Ulrich Menne for alerting us on the relation between JLT2020 and (Federer1969, , 2.8.9). We appreciate Duc Hoang Luu for Example 1 (3) and helpful suggestions. We are grateful to the anonymous referee for critical comments and valuable suggestions. HVL would like to thank Domenico Fiorenza and XuanLong Nguyen for helpful comments on the first version of this paper Le2019 . She warmly thanks JJ, TDT and Duc Hoang Luu for hospitality during her visit to MPI MIS Leipzig in 2019 where a part of this paper was discussed.